Effective action approach to higher-order relativistictidal interactions in binary systems and their effective

one body description

Donato BINI, Thibault DAMOUR and Guillaume FAYE

Institut des Hautes Etudes Scientifiques

35, route de Chartres

91440 – Bures-sur-Yvette (France)

Juin 2012

IHES/P/12/10

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Effective action approach to higher-order relativistic tidal interactions in binary

systems and their effective one body description

Donato BiniIstituto per le Applicazioni del Calcolo “M. Picone,” CNR, I-00185 Rome, Italy

Thibault DamourInstitut des Hautes Etudes Scientifiques, F-91440 Bures-sur-Yvette, France

Guillaume FayeInstitut d’Astrophysique de Paris, UMR 7095 CNRS,

Universite Pierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France(Dated: June 18, 2012)

The gravitational-wave signal from inspiralling neutron-star–neutron-star (or black-hole–neutron-star) binaries will be influenced by tidal coupling in the system. An important science goal in thegravitational-wave detection of these systems is to obtain information about the equation of stateof neutron star matter via the measurement of the tidal polarizability parameters of neutron stars.To extract this piece of information will require accurate analytical descriptions both of the motionand the radiation of tidally interacting binaries. We improve the analytical description of the lateinspiral dynamics by computing the next-to-next-to-leading order relativistic correction to the tidalinteraction energy. Our calculation is based on an effective-action approach to tidal interactions, andon its transcription within the effective-one-body formalism. We find that second-order relativisticeffects (quadratic in the relativistic gravitational potential u = G(m1 + m2)/(c

2r)) significantlyincrease the effective tidal polarizability of neutron stars by a distance-dependent amplificationfactor of the form 1 + α1 u + α2 u2 + · · · where, say for an equal-mass binary, α1 = 5/4 = 1.25 (aspreviously known) and α2 = 85/14 ≃ 6.07143 (as determined here for the first time). We arguethat higher-order relativistic effects will lead to further amplification, and we suggest a Pade-typeway of resumming them. We recommend testing our results by comparing resolution-extrapolatednumerical simulations of inspiralling-binary neutron stars to their effective one body description.

PACS numbers: 04.30.-w, 04.25.Nx

Keywords:

I. INTRODUCTION

Inspiralling binary neutron stars are among the mostpromising sources for the advanced versions of the cur-rently operating ground-based gravitational-wave (GW)detectors LIGO/Virgo/GEO. These detectors will bemaximally sensitive to the inspiral part of the GW signal,which will be influenced by tidal interaction between twoneutron stars. An important science goal in the detectionof these systems (and of the related mixed black-hole–neutron-star systems) is to obtain information about theequation of state of neutron-star matter via the mea-surement of the tidal polarizability parameters of neu-tron stars. The analytical description of tidally interact-ing compact-binary systems (made of two neutron starsor one black hole and one neutron star) has been initi-ated quite recently [1–8]. In addition, these analyticaldescriptions have been compared to accurate numericalsimulations [5, 9–11], and have been used to estimatethe sensitivity of GW signals to the tidal polarizabilityparameters [11–15].

Here, we shall focus on one aspect of the analyticaldescription of tidally interacting relativistic binary sys-tems, namely the role of the higher-order post-Newtonian(PN) corrections in the tidal interaction energy, as de-

scribed, in particular, within the effective one body(EOB) formalism [16–19]. Indeed, the analysis of Ref. [5],which compared the prediction of the EOB formalismfor the binding energy of tidally interacting neutronstars to (nonconformally flat) numerical simulations ofquasi-equilibrium circular sequences of binary neutronstars [20, 21], suggested the importance of higher-orderPN corrections to tidal effects, beyond the first post-Newtonian (1PN) level, and their tendency to signifi-

cantly increase the “effective tidal polarizability” of neu-tron stars.

In the EOB formalism, the gravitational binding of abinary system is essentially described by a certain “radialpotential” A(r). In the tidal generalization of the EOBformalism proposed in Ref. [5], the EOB radial potentialA(r) is written as the sum of three contributions,

A(r) = ABBH(r) +AtidalA (r) +Atidal

B (r) , (1.1)

where ABBH(r) is the radial potential describing the dy-namics of binary black holes, and where Atidal

A (r) andAtidal

B (r) are the additional radial potentials associated,respectively, with the tidal deformations of body A andbody B. [For binary neutron-star systems both Atidal

A

and AtidalB are present, while for mixed neutron-star–

black-hole systems only one term, corresponding to the

2

neutron star, is present; see following]. Here, we considera binary system of (gravitational) masses mA and mB,and denote

M ≡ mA +mB , ν ≡ mAmB

(mA +mB)2. (1.2)

[A labelling of the two bodies by the letters A and B willbe used in this Introduction for writing general formulas.We shall later use the alternative labelling A = 1, B = 2when explicitly dealing with the metric generated by thetwo bodies.] The binary black-hole (or point mass) po-tential ABBH(r) is known up to the third post-Newtonian(3PN) level [18], namely

ABBH3PN (r) = 1− 2 u+ 2 ν u3 + a4 ν u

4 , (1.3)

where a4 = 94/3− (41/32)π2 ≃ 18.68790269, and

u ≡ GM

c2 r, (1.4)

with c being the speed of light in vacuum and G theNewtonian constant of gravitation.

It was recently found [22, 23] that an excellent descrip-tion of the dynamics of binary black-hole systems is ob-tained by augmenting the 3PN expansion Eq. (1.3) withadditional fourth post-Newtonian (4PN) and fifth post-Newtonian (5PN) terms, and by Pade resumming thecorresponding 5PN Taylor expansion.

The tidal contributions AtidalA,B (r) can be decomposed

according to multipolar order ℓ, and type, as

AtidalA (r) =

∑

ℓ≥2

A

(ℓ) LOA electric(r) A

(ℓ)A electric(r)

+A(ℓ) LOA magnetic(r) A

(ℓ)A magnetic(r) + . . .

. (1.5)

Here, the label “electric” refers to the gravito-electrictidal polarization induced in body A by the tidal fieldgenerated by its companion, while the label “magnetic”refers to a corresponding gravito-magnetic tidal polar-ization. On the other hand, the label LO refers to theleading-order approximation (in powers of u) of each(electric or magnetic) multipolar radial potential. Forinstance, the gravito-electric contribution at multipolarorder ℓ is equal to [5]

A(ℓ) LOA electric(r) = −κ(ℓ)

A u2ℓ+2 (1.6)

where

κ(ℓ)A = 2 k

(ℓ)A

mB

mA

(RA c

2

G(mA +mB)

)2ℓ+1

. (1.7)

Here, RA denotes the radius of body A, and k(ℓ)A de-

notes a dimensionless “tidal Love number”. [Note that

k(ℓ)A was denoted kA

ℓ in our previous work. Here weshall always put the multipolar index ℓ within paren-theses to avoid ambiguity with our later use of the la-belling A,B = 1, 2 for the two bodies.] The correspond-ing leading-order radial potential of the gravito-magnetic

type is proportional to u2ℓ+3 (instead of u2ℓ+2), and

to j(ℓ)A R2ℓ+1

A , where j(ℓ)A denotes a dimensionless “mag-

netic tidal Love number”. It was found [3, 4] that bothtypes of Love numbers have a strong dependence uponthe compactness CA ≡ GmA/(c

2RA) of the tidally de-

formed body, and that both k(ℓ)A and j

(ℓ)A contain a factor

1− 2 CA, so that they would formally vanish in the limitwhere body A becomes as compact as a black hole (i.e.CA → CBH = 1

2 ). This is consistent with the decomposi-tion Eq. (1.1), where the binary black-hole radial poten-tial ABBH(r) is the only remaining contribution when oneformally takes the limit where both CA and CB tend tothe black-hole value CBH = 1/2. Finally, the supplemen-

tary factors A(ℓ)A electric(r) and A

(ℓ)A magnetic(r) denote the

distance-dependent amplification factors of the leading-order tidal interaction by higher-order PN effects. Theyhave the general form

A(ℓ)A electric(r) = 1 + α

A(ℓ)1 electricu+ α

A(ℓ)2 electricu

2 + . . . , (1.8)

A(ℓ)A magnetic(r) = 1 + α

A(ℓ)1 magneticu+ . . . , (1.9)

where u is defined by Eq. (1.4).

The main aim of the present investigation will be to

compute the electric-type amplification factors A(ℓ)A electric,

for ℓ = 2 (quadrupolar tide) and ℓ = 3 (octupolar tide),

at the second order in u, i.e. to compute both αA(ℓ)1 electric

and αA(ℓ)2 electric. We shall also compute the magnetic-type

amplification factor A(ℓ)A magnetic, for ℓ = 2, at the first

order in u.

The analytical value of the first-order electric amplifi-

cation coefficient αA(ℓ)1 electric was computed some time ago

for ℓ = 2 (see Ref. [29] in [5]) and was reported in Eq. (38)of [5], namely

αA(ℓ=2)1 electric =

5

2XA , (1.10)

where XA ≡ mA/(mA + mB) is the mass fraction ofbody A. The analytical result (1.10) has been recentlyconfirmed [6]. On the other hand, several comparisons ofthe analytical description of tidal effects with the resultsof numerical simulations have indicated that the ampli-

fication factor A(ℓ=2)A electric(r) is larger that its 1PN value

1 +αA(ℓ=2)1 electric u, and have suggested that the higher-order

coefficients αA(ℓ)2 electric, . . . take large, positive values. More

precisely, the analysis of Ref. [5] suggested (when takinginto account the value (1.10) for α1) a value of order

αA(ℓ=2)2 electric ∼ + 40 (for the equal-mass case) from a com-

parison with the numerical results of Refs. [20, 21] onquasi-equilibrium adiabatic sequences of binary neutronstars. Recently, a comparison with dynamical simula-tions of inspiralling binary neutron stars confirmed theneed for such a large value of αA

2 electric [9, 10]. [Notethat, while the comparison to the highest resolution nu-merical data suggests the need of even larger values of

3

αA(ℓ=2)2 electric, of order + 100, the comparison to approximate

resolution-extrapolated data call only for α2 values of or-der + 40. See Fig. 13 in [10].]

II. EFFECTIVE ACTION APPROACH TOTIDAL EFFECTS

A. Finite-size effects and nonminimal worldlinecouplings

It was shown long ago [24], using the technique ofmatched asymptotic expansions, that the motion and ra-diation of N (non-spinning) compact objects can be de-scribed, up to the 5PN approximation, by an effectiveaction of the type

S0 =

∫dDx

c

c4

16 πG

√g R(g) + Spointmass , (2.1)

where R(g) represents the scalar curvature associatedwith the metric gµν , with determinant −g, and where

Spointmass = −∑

A

∫mA c

2 dτA (2.2)

is the leading order skeletonized description of the com-pact objects, as point masses. Here dτA denotes theproper time along the worldline yµ

A(τA) of A, namely

dτA ≡ c−1(−gµν(yA) dyµA dy

νA)1/2. To give meaning to

the notion of point mass sources in General Relativityone needs to use a covariant regularization method. Themost convenient one is dimensional regularization, i.e.analytic continuation in the value of the spacetime di-mension D = 4 + ε, with ε ∈ C being continued to zeroonly at the end of the calculation. The consistency andefficiency of this method has been shown in the calcula-tions of the motion [25, 26] and radiation [27] of binaryblack holes at the 3PN approximation.

It was also pointed out in Ref. [24] that finite-sizeeffects (linked to tidal effects, and the fact that neu-tron stars have, contrary to black holes, non-zero Love

numbers k(ℓ)A ) enter at the 5PN level. In effective field

theory, finite-size effects are treated by augmenting thepoint-mass action of Eq. (2.2) by nonminimal world-line couplings involving higher-order derivatives of thefield [28–30]. In a gravitational context this meansconsidering worldline couplings involving the 4-velocityuµ

A ≡ dyµA/dτA (satisfying gµν u

µA u

νA = −c2) together

with the Riemann tensor Rαβµν and its covariant deriva-tives. To classify the possible worldline scalars that canbe constructed one can appeal to the relativistic theory oftidal expansions [31–33]. In the notation of Refs. [32, 33]the tidal expansion of the “external metric” felt by bodyA can be entirely expressed in terms of two types of exter-nal tidal gradients evaluated along the central worldlineof this body: the gravito-electric GA

L(τA) ≡ GAa1...aℓ

(τA)

and gravito-magnetic HAL (τA) ≡ HA

a1...aℓ(τA) symmet-

ric trace-free (spatial) tensors, together with their time-derivatives. [The spatial indices ai = 1, 2, 3 refer to alocal frame X0

A ≡ c τA, XaA attached to body A.] This

implies that the most general nonminimal worldline ac-tion has the form

Snonminimal =∑

A

∑

ℓ≥2

1

2

1

ℓ!µ

(ℓ)A

∫dτA(GA

L(τA))2

+1

2

ℓ

ℓ+ 1

1

ℓ!

1

c2σ

(ℓ)A

∫dτA(HA

L (τA))2

+1

2

1

ℓ!

1

c2µ′(ℓ)A

∫dτA(GA

L(τA))2

+1

2

ℓ

ℓ+ 1

1

ℓ!

1

c4σ′(ℓ)A

∫dτA(HA

L (τA))2

+ . . .

, (2.3)

where GAL(τA) ≡ dGA

L/dτA, and where the ellipsis refereither to higher proper-time derivatives ofGA

L andHAL , or

to higher-than-quadratic invariant monomials made fromGA

L , HAL and their proper-time derivatives. For instance,

the leading-order non-quadratic term would be∫dτA GA

ab GAbc G

Aca . (2.4)

Note that the allowed monomials in GL, HL and theirtime derivatives are restricted by symmetry constraints.When considering a non-spinning neutron star (which issymmetric under time and space reflections) one shouldonly allow monomials invariant under time and space re-versals. For instance Gab Gab and GabHab are not al-lowed.

B. Tidal coefficients

The electric-type tidal moments GAL are normalized

in a Newtonian way, i.e. such that, in lowest PN or-der, they reduce to the usual Newtonian tidal gradi-ents: GA

L = [∂L U(Xa)]Xa=0 + O(

1c2

), where U(X) is

the Newtonian potential and ∂L ≡ ∂i1∂i2 . . . ∂iℓrepre-

sents multiple ordinary space derivatives. The magnetic-type ones HA

L are defined (in lowest PN order) as re-peated gradients of the gravitomagnetic field c3g0a. With

these normalizations the coefficients µ(ℓ)A and σ

(ℓ)A in the

nonminimal action in Eq. (2.3) both have dimensions[length]2ℓ+1/G. They are related to the dimensionless

Love numbers k(ℓ)A and j

(ℓ)A , and to the radius of body A,

via [3]

Gµ(ℓ)A =

1

(2ℓ− 1)!!2 k

(ℓ)A R2ℓ+1

A , (2.5)

Gσ(ℓ)A =

ℓ− 1

4(ℓ+ 2)

1

(2ℓ− 1)!!j(ℓ)A R2ℓ+1

A . (2.6)

4

Note that the coefficients associated with the first timederivatives of GA

L and HAL have dimensions Gµ

′(ℓ)A ∼

[length]2ℓ+3 ∼ Gσ′(ℓ)A . The nonminimal action in Eq.

(2.3) has a double ordering in powers of RA and in pow-ers of 1/c2. The lowest-order terms in the RA expansionare proportional to R5

A and correspond to the electricand magnetic quadrupolar tides, as measured by GA

ab andHA

ab, respectively.

C. Tidal tensors

We have written the most general nonminimal actionEq. (2.3) in terms of the irreducible symmetric trace-freespatial tensors [with respect to the local space associatedwith the worldline yµ

A(τA)] describing the tidal expan-sion of the “external metric” felt by body A, as definedin Ref. [32]. These tidal tensors played a useful role insimplifying the (1PN-accurate) relativistic theory of tidaleffects. In our present investigation, it will be convenientto express them in terms of the Riemann tensor and itscovariant derivatives. Eq. (3.40) in Ref. [32] shows that(in the case where one can neglect corrections propor-tional to the covariant acceleration of the worldline) thefirst two electric spatial tidal tensors, Gab and Gabc, aresimply equal (modulo a sign) to the non-vanishing spatialcomponents (in the local frame) of the following space-time tensors (evaluated along the considered worldline)

Gαβ ≡ −Rαµβν uµ uν , (2.7)

Gαβγ ≡ − Symαβγ(∇⊥α Rβµγν)uµ uν . (2.8)

Here the notation Gαβ for (minus) the electric partof the curvature tensor should not be confused with theEinstein tensor, Symαβγ denotes a symmetrization (with

weight one) over the indices αβ γ, while ∇⊥α ≡ P (u)µ

α∇µ

denotes the projection of the spacetime gradient ∇µ or-thogonally to uµ (P (u)µ

ν ≡ δµν + c−2 uµ uν). [Note that

in the Newtonian limit uµ ≃ c δµ0 so that the Newto-

nian limit of Gαβ is − c2Rα0β0, where the factor c2 can-cels the O(1/c2) order of the curvature tensor.] By con-trast, the presence of the extra term − 3 c−2E∗

〈aE∗b〉 on

the right-hand side of Eq. (3.40) in Ref. [32] showsthat the ℓ = 4 electric spatial tidal tensor Gabcd =∂〈abc E

∗d〉 would differ from the symmetrized spatial pro-

jection of (∇α∇β Rγµδν)uµ uν by a term proportional toG〈αγ Gβδ〉. (Here, the angular brackets denote a (spatial)symmetric trace-free projection.) In addition, the electric

time derivatives, such as Gab can be replaced by corre-sponding spacetime tensors such as uµ∇µGαβ . Similarlyto Eqs. (2.7), (2.8), one finds that the ℓ = 2 and ℓ = 3magnetic tidal tensors (as defined in Refs. [32, 33]) areequal to the nonvanishing local-frame spatial componentsof the spacetime tensors

Hαβ ≡ + 2 cR∗αµβν uµ uν , (2.9)

Hαβγ ≡ + 2 c Symαβγ(∇⊥α R

∗βµγν)uµ uν , (2.10)

where R∗µναβ ≡ 12 ǫµνρσ R

ρσαβ is the dual of the curva-

ture tensor, ǫµνρσ denoting here the Levi-Civita tensor(with ǫ0123 = +

√g). Note the factor +2 entering the

link between the magnetic tidal tensorsHαβ , . . . (normal-ized as in Refs. [32, 33]) and the dual of the curvaturetensor, which contrasts with the factor −1 entering thecorresponding electric tidal-tensor links, Eqs. (2.7), (2.8).(The definition of BA

αβ in the text below Eq. (5) of Ref.

[5] should have included such a factor 2 in its right-handside. On the other hand, the corresponding magnetic-quadrupole tidal action, Eq. (13) there, was computedwith Hab and was correctly normalized.) Let us also notethat the expressions in Eqs. (2.7)–(2.10) assume that theRicci tensor vanishes (e.g. to ensure the tracelessness ofGαβ). One could have, alternatively, defined Gαβ etc. byusing the Weyl tensor Cαµβν instead of Rαµβν . However,as discussed in Ref. [29], the terms in an effective actionwhich are proportional to the (unperturbed) equations ofmotion (such as Ricci terms) can be eliminated (modulocontact terms) by suitable field redefinitions.

D. Covariant description of tidal interactions

Finally, the covariant form of the effective action de-scribing tidal interactions reads

Stot = S0 + Spointmass + Snonminimal (2.11)

where S0 and Spointmass are given by Eqs. (2.1), (2.2),and where the covariant form of the nonminimal world-line couplings starts as

Snonminimal =∑

A

1

4µ

(2)A

∫dτAG

Aαβ G

αβA

+1

6 c2σ

(2)A

∫dτAH

Aαβ H

αβA

+1

12µ

(3)A

∫dτA G

Aαβγ G

αβγA

+1

4 c2µ′(2)A

∫dτA(uµ

A∇µGAαβ)(uν

A∇νGαβA )

+ . . .

, (2.12)

where GαβA ≡ gαµ gβν GA

µν , etc. [evaluated along the Aworldline].

In principle, one can then derive the influence of tidalinteraction on the motion and radiation of binary sys-tems by solving the equations of motion following fromthe action of Eqs. (2.11), (2.12). More precisely, thisaction implies both a dynamics for the worldlines wherethe geodesic equation is modified by tidal forces [com-ing from δSnonminimal/δ y

µA(τA)], and modified Einstein

5

equations for the gravitational field of the type

Rµν −1

2Rgµν =

8πG

c4T pointmass

µν + T nonminimalµν

,

(2.13)where the new tidal sources T µν

nonminimal(x) =

(2c/√g) δSnonminimal/δ gµν(x) are, essentially, sums

of derivatives of worldline Dirac-distributions:

Tnonminimal(x) ∼∑

A

∑

ℓ

∂ℓ δ(x− yA) .

E. A simplifying, general property of reduced actions

The task of solving the coupled dynamics of the world-lines and of the gravitational field, both being modifiedby tidal effects, at the second post-Newtonian (2PN)level, i.e. at the next-to-next-to-leading order in tidaleffects, and then of computing the looked for higher-order terms in the amplification factors of Eqs. (1.8),(1.9) is quite non-trivial. Happily, one can drasticallysimplify the needed work by using a general property ofreduced actions. Indeed, we are interested here in know-ing the influence of tidal effects on the reduced dynam-ics of a compact binary, that is, the dynamics of thetwo worldlines yµ

A(τ), yµB(τ), obtained after having “in-

tegrated out” the gravitational field (i.e., after havingexplicitly solved gµν(x) as a functional of the two world-lines). When considering, as we do here, the conservative

dynamics of the system (without radiation reaction), itcan be obtained from a reduced action, which is tradi-tionally called the “Fokker action”. See Ref. [28] andreferences therein for a detailed discussion (using a dia-grammatic approach) of Fokker actions (at the 2PN level,and with the inclusion of scalar couplings in addition tothe pure Einsteinian tensor couplings). If we denote thefields mediating the interaction between the worldlinesy = yA, yB as ϕ (in our case ϕ = gµν), the reducedworldline action Sred[y] (a functional of the worldlines y)that corresponds to the complete action S[ϕ, y] describ-ing the coupled dynamics of y and ϕ is formally definedas:

Sred [y] ≡ S [ϕsol [y], y] , (2.14)

where ϕsol [y] is the functional of y obtained by solvingthe ϕ-field equation,

δ S [ϕ, y]/δϕ = 0 , (2.15)

considered as an equation for ϕ, with given source-worldlines. (This must be done with time-symmetricboundary conditions and, in the case of gµν , the addi-tion of a suitable gauge-fixing term; see Ref. [28] fordetails.)

Having recalled the concept of reduced (or Fokker) ac-tion, let us now consider the case where the completeaction is of the form

S [ϕ, y] = S(0)[ϕ, y] + ǫ S(1)[ϕ, y] , (2.16)

where ǫ denotes a “small parameter”. In our case, ǫcan be either a formal parameter associated with allthe nonminimal tidal terms in Snonminimal, Eq. (2.12),or, more concretely, any of the tidal parameters entering

Eq. (2.12): µ(ℓ=2)A , µ

(ℓ=2)B , etc. As said previously, when

turning on ǫ, the equations of motion, and therefore thesolutions of both ϕ and y get perturbed by terms of or-der ǫ : ϕ = ϕ(0) + ǫ ϕ(1) + . . ., y = y(0) + ǫ y(1) + . . ., buta simplification occurs when considering the reduced ac-tion Eq. (2.14). Indeed, it is true that the field equation(2.15) for ϕ gets modified into

0 =δ S [ϕ, y]

δ ϕ=δ S(0)[ϕ, y]

δ ϕ+ ǫ

δ S(1)[ϕ, y]

δ ϕ, (2.17)

so that its solution ϕsol [y] gets perturbed:

ϕsol [y] = ϕ(0)sol [y] + ǫ ϕ

(1)sol [y] +O(ǫ2) . (2.18)

However, when inserting the perturbed solution of Eq.(2.18) into the complete, perturbed action of Eq. (2.16),one finds

Sred [y] = S [ϕ(0)sol [y] + ǫ ϕ

(1)sol [y] +O(ǫ2), y]

= S [ϕ(0)sol [y], y] + ǫ ϕ

(1)sol [y]

δ S

δ ϕ[ϕ

(0)sol [y], y] +O(ǫ2)

= S [ϕ(0)sol [y], y]

+ǫ ϕ(1)sol [y]

δ S(0)

δ ϕ[ϕ

(0)sol [y], y] +O(ǫ2)

= S [ϕ(0)sol [y], y] +O(ǫ2) , (2.19)

because, by definition, ϕ(0)sol is a solution of δ S(0)/δ ϕ = 0.

Note that, in Eq. (2.19), while the functional S is thecomplete, perturbed action, the functional argument isthe unperturbed solution. Decomposing the functional Sinto its unperturbed plus perturbed parts [see Eq. (2.16)]then leads to the final result:

Sred [y] = S(0)[ϕ(0)sol [y], y] + ǫ S(1)[ϕ

(0)sol [y], y] +O(ǫ2)

= S(0)red [y] + ǫ S(1)[ϕ

(0)sol [y], y] +O(ǫ2) . (2.20)

In words: the order O(ǫ) perturbation

ǫ S(1)red [y] ≡ Sred [y]− S

(0)red [y]

of the reduced action is correctly obtained, modulo termsof order O(ǫ2), by replacing in the O(ǫ) perturbation

ǫ S(1) [ϕ, y]

of the complete (unreduced) action the field ϕ by its un-

perturbed solution ϕ(0)sol [y].

In our case, the ordering parameter ǫ is either the col-

lection µ(2)A , µ

(2)B , µ

(3)A , µ

(3)B , . . . , σ

(2)A c−2, . . . , µ

′(2)A c−2, . . .,

or the corresponding sequence of powers of RA andRB : R5

A, R5B , R

7A, R

7B, . . . The terms quadratic in ǫ would

6

therefore involve at least ten powers of the radii (andwould mix with higher-than-quadratic worldline contri-butions akin to (2.4)). Neglecting such terms, we con-clude that the higher-PN corrections to the tidal effectsare correctly obtained by replacing in Eq. (2.12), con-sidered as a functional of gµν(x) and yµ

A(τA), the metricgµν(x) by the point-mass metric obtained by solving Ein-stein’s equations with point-mass sources. [This was themethod used by one of us (T.D.) to compute the 1PNcoefficient of Eq. (1.10) from the calculation by Damour,Soffel and Xu of the 1PN-accurate value of Gab [34, 35].]

III. THE 2PN POINT-MASS METRIC AND ITSREGULARIZATION

A. Form of the 2PN point-mass metric

The result of the last Section allows one to computethe tidal corrections to the reduced action for two tidallyinteracting bodies A,B with the same accuracy at whichone knows the metric generated by two (structureless)point masses mA, y

µA;mB, y

µB. The metric generated by

two point masses has been the topic of many works overmany years. It has been known (in various forms andgauges) at the 2PN approximation for a long time [36–38]. Here, we shall use the convenient, explicit harmonic-gauge form of Ref. [39], with respect to the (harmonic)coordinates xµ = (x0 ≡ ct, xi), i.e. the metric

ds2 = g00(dx0)2 + 2 g0i dx

0dxi + gij dxidxj , (3.1)

where, at 2PN, the metric components are written as

g00 = −1 + 2 ǫ2 V − 2 ǫ4 V 2

+ 8 ǫ6(X + δij ViVj +

1

6V 3

)+O(8) ,

g0i = − 4 ǫ3 Vi − 8 ǫ5Ri +O(7) ,

gij = δij(1 + 2 ǫ2 V + 2 ǫ4 V 2

)+ 4 ǫ4 Wij +O(6) .

(3.2)

Here, as below, we sometimes use the alternative notationǫ ≡ 1/c for the small PN parameter. We used also theshorthand notation O(n) ≡ O(ǫn) ≡ O(c−n).

The various 2PN brick potentials V, Vi, Wij , Ri and Xare the (time-symmetric) solutions of

V = − 4 πGσ ,

Vi = − 4 πGσi ,

Wij = − 4 πG(σij − δij σkk)− ∂iV ∂jV ,

Ri = − 4 πG(V σi − Viσ)− 2 ∂kV ∂iVk −3

2∂tV ∂iV ,

X = − 4 πGV σii + 2Vi ∂t ∂iV + V ∂2t V +

3

2(∂tV )2

− 2 ∂iVj ∂jVi + Wij ∂ijV , (3.3)

where ∂t denotes a time derivative (while we remind that∂i, for instance, denotes a spatial one), and where thecompact-supported source terms are [40]

σ ≡ T 00 + T ii

c2, σi ≡

T 0i

c, σij ≡ T ij , (3.4)

with T µν being the stress-energy tensor of two pointmasses:

T µν = µ1(t) vµ1 (t) vν

1 (t) δ(x− y1(t)) + 1 ↔ 2 , (3.5)

where

µ1(t) = m1

[g−1/2(gµν v

µ1 v

ν1/c

2)−1/2]1. (3.6)

Here, vµ1 =

dyµ1

dt = (c, vi1) and the index 1 on the bracket

in Eq. (3.6) refers to a regularized limit where the fieldpoint xi tends towards the (point-mass) source point yi

1.Note that, in this section, we shall generally label thetwo particles as (m1, y

i1), (m2, y

i2), instead of (mA, y

iA),

(mB, yiB) as above. The notation 1 ↔ 2 means adding

the terms obtained by exchanging the particle labels 1and 2.

The explicit forms of the 2PN-accurate brick potentialsV , Vi, Wij , Ri, X were given in Ref. [39]. Their time-symmetric parts are recalled in Appendix A. These brickpotentials are expressed as explicit functions of r1 ≡ x−y1, r1 ≡ |r1|, n1 ≡ r1/r1, r2 ≡ x − y2, etc., y12 ≡y1 − y2, r12 ≡ |y12|, n12 ≡ y12/r12, v12 ≡ v1 − v2,(n12 v1) ≡ n12 · v1. Note the appearance of the auxiliaryquantity S, which denotes the perimeter of the triangledefined by x, y1 and y2, viz

S ≡ r1 + r2 + r12 . (3.7)

In all the PN expressions, the spacetime pointsxµ, yµ

1 , yµ2 (and the velocities vµ

A) are taken at the sameinstant t, i.e. x0 = y0

1 = y02 = ct.

B. Regularization of the 2PN metric and of the2PN tidal actions

Let us now discuss in more detail the crucial oper-ation (already implicit in Sec. II above) of regulariza-tion of all the needed field quantities, such as gµν(x),g(x), Rµανβ(x), . . ., when they are to be evaluated on aworldline: xµ → yµ

A. As mentionned at the beginning ofSec. II, all the quantities [Gµν(x)]1, . . . , [Rµανβ(x)]1 aredefined by dimensional continuation. It was shown longago [24, 41] that, at 2PN, dimensional regularization isequivalent to the Riesz’ analytic regularization, and isa technical shortcut for computing the physical answerobtained by the matching of asymptotic expansions. Inaddition, because of the restricted type of singular termsthat appear at 2PN [see Eqs. (25), (30) and (33) in Ref.[24]], the analytic-continuation regularization turns outto be equivalent to Hadamard regularization (used, at

7

2PN, in Refs. [38, 39, 42]); see below. Here, it will betechnically convenient to use Hadamard regularization(which is defined in D = 4) because the explicit form ofEqs. (A1)–(A5) of the 2PN metric that we shall use ap-plies only in the physical dimension D = 4 and has lostthe information about its dimensionally continued kin inD = 4 + ε.

Let us summarize here the (Hadamard-type) defini-tion of the regular part of any field quantity ϕ(x) (whichmight be a brick potential, V (x), Vi(x), . . ., a componentof the metric gµν(x), or a specific contribution to a tidalmoment, Gαβ , . . .). We consider the behavior of ϕ(x)near particle 1, i.e. when r1 = |x − y1| → 0. To easethe notation, we shall provisionally put the origin of the(harmonic) coordinate system at y1 (at some instant t),i.e. we shall assume that y1 = 0, so that r1 = |x| ≡ rand n1 = r1/r1 = x/r ≡ n. We consider the expan-sion of ϕ(x) in (positive and negative) integer powers kof r1 = r, and in spherical harmonics of the directionn1 = n, say (for k ∈ Z, ℓ ∈ N, N ∈ N)

ϕ(x) =∑

k≥−N

∑

ℓ≥0

rk nL fkL , (3.8)

where nL ≡ na1...aℓ denotes the symmetric trace-free pro-jection of the tensor nL ≡ na1 . . . naℓ . [The angular func-

tion fkL n

L is equivalent to a sum of+ℓ∑

m=−ℓ

cm Yℓm.] We

(uniquely) decompose the field ϕ(x) in a regular part (R)and a singular one (S),

ϕ(x) = R [ϕ(x)] + S [ϕ(x)] , (3.9)

by defining (n ∈ N)

R [ϕ(x)] ≡∑

ℓ≥0

∑

n≥0

rℓ+2n nLf ℓ+2nL , (3.10)

S [ϕ(x)] ≡∑

k 6=ℓ+2n

rk nLfkL . (3.11)

Note that R [ϕ(x)] can be rewritten as a sum of infinitelydifferentiable terms of the type xL(x2)n. By contrastS [ϕ(x)] is such that it (if N , in Eq. (3.8), is strictlypositive), or, one of its (repeated) spatial derivatives,tends towards infinity as r → 0. Note also that theR+S decomposition commutes with linear combinations(with constant coefficients), as well as with spatial deriva-tives, in the sense that R [aϕ(x)+ b ψ(x)] = aR [ϕ(x)]+bR [ψ(x)], S [aϕ(x) + b ψ(x)] = aS [ϕ(x)] + b S [ψ(x)],R [∂i ϕ(x)] = ∂i R [ϕ(x)] and S [∂i ϕ(x)] = ∂i S [ϕ(x)].By contrast, the R+S decomposition (as defined above,in the Hadamard way) does not commute with nonlin-ear operations (e.g. R [ϕψ] 6= R [ϕ]R [ψ]), nor evenwith multiplication by a smooth (C∞) function f(x) (e.g.R [fϕ] 6= f R [ϕ]). This is a well-known inconsistency ofthe Hadamard regularization, which created many am-biguities when it was used at the 3PN level [43, 44].

One might worry that our present calculation (whichaims at regularizing nonlinear quantities quadratic inRµανβ ∼ ∂2g+ g−1 ∂g ∂g) might be intrinsically ambigu-ous already at the 2PN level. Actually, this turns outnot to be the case because of the special structure ofthe 2PN metric which is at work in the Riesz-analytic-continuation derivation of the 2PN dynamics in Ref. [24].This structure guarantees, in particular, that the Rie-mann tensor (or its derivatives) is regularized unambigu-ously.

C. On the special structure of the 2PN metricguaranteeing its unambiguous regularization

Let us first recall why the Riesz-analytic-continuationmethod, or, equivalently (when considering the reg-ularization of the metric and its derivatives), thedimensional-continuation method, is consistent undernonlinear operations. The dimensional-continuation ana-log of Eqs. (3.9)–(3.11) consists of distinguishing, withinϕ(x), the terms that (in dimension 4 + ε) contain pow-ers of r of the type rk−nε, with n = 1, 2, 3, . . . [whichdefine the ε-singular part of ϕ(x)], and the terms thatare (formally) C∞ in 4 + ε dimensions [which definethe ε-regular part of ϕ(x)]. It is then easily seen in di-mensional continuation (simply by considering the con-tinuation to large, negative values of the real part ofε) that the ε-singular terms give vanishing contribu-tions when evaluated at r → 0, and that they do soconsistently in nonlinear terms such as ∂ϕ∂ψ. Letus now indicate why the special structure of the 2PNmetric ensures that the decomposition into ε-singularparts and ε-regular parts of the various brick potentialsV (x), Vi(x), . . . coincides with their above-defined decom-position into Hadamard-singular (S [V (x)], S [Vi(x)], . . .)and Hadamard-regular parts (R [V (x)], R [Vi(x)], . . .) inthe four-dimensional case. This is trivially seen to bethe case for most of the 2PN contributions to the brickpotentials (because one easily sees how those contribu-tions smoothly evolve when analytically continuing thedimension). However, the most nonlinear contributions

to the 2PN metric, namely the terms, say X(V V V ), in Xthat are generated by the cubically nonlinear terms con-

tained in the last source term, W(V V )ij ∂ijV , on the right

hand-side of the last Eq. (3.3) (where W(V V )ij is the part

of Wij generated by −∂iV ∂jV ) are more delicate to dis-

cuss. Actually, among the contribution X(V V V ), only theterms proportional either to m2

1m2 or to m1m22, i.e., the

terms whose cubically nonlinear source ∼ ∂2V∆−1∂V ∂Vinvolve two V potentials generated by one worldline andone V potential generated by the other worldline, such asX(V1V1V2) ∝ m2

1m2, pose a somewhat delicate problem.More precisely, it is easily seen that the only danger-ous part in X(V1V1V2), considered near the first world-

line, is of the form f(x)/r(2+2ε)1 in dimension 4 + ε,

where f(x) denotes a smooth function. [Here, we add

8

back the particle label indicating whether the expan-sions Eqs. (3.10), (3.11) refer to the first (A = 1),or the second (A = 2) particle. The appropriate la-bel should be added both on r and n in Eqs. (3.10),(3.11): rk nL → rk

A nLA.] The problem is that the power

of 1/r1 in this ε-singular term becomes an even inte-

ger when ε → 0. When inserting the Taylor expan-sion of f(x), say f(x) ∼ ∑ rℓ+2n

1 nL1 f

ℓ+2nL , some of the

terms in the ε-singular contribution f(x)/r(2+2ε)1 might

be of the form rℓ+2n′−2ε1 nL

1 , with n′ = n − 1 ≥ 0, andmight then contribute to the Hadamard-regular part ofX(V1V1V2) in the limit ε → 0. This would mean thatthe Hadamard-regular part of X(V1V1V2) would not co-incide with its ε-regular part. We already know fromRefs. [38, 39]), which used Hadamard regularization toderive the 2PN-accurate dynamics and found the sameresult (modulo gauge effects) as the analytic-continuationderivation of Ref. [24], that this is not the case for the

regularized values of X(V1V1V2) and of its first derivativeson the first worldline. [Indeed, these quantities enterthe computation of the equations of motion.] On theother hand, the computations that we shall do here in-volve higher spatial derivatives of X, and it is importantto check that we can safely use Hadamard regulariza-tion to evaluate them. This can be proven by using thetechniques explained in Ref. [24], based on iteratively

considering the singular terms in W(V V )ij and X(V V V )

generated by the singular local behaviour (near the firstworldline) of their respective source terms. One findsthen that the smooth function f(x) entering the dan-

gerous terms f(x)/r(2+2ε)1 in X(V1V1V2) is of the special

form f(x) ∼∑cℓGLr

ℓ1 n

L1 in dimension 4+ε, with ℓ ≥ 1,

where GL ≡ ∂LV2 denotes the ℓ-th tidal gradient (consid-ered near the first worldline) of the V potential generatedby the second worldline. When working (as we do) at the2PN accuracy, we can take V at Newtonian order, and

the gradients GL ≃ [∂L(Gm2/r(1+ε)2 )]1 are then trace-

less: GL = Ga1a2···aℓ= G〈a1a2···aℓ〉. As a consequence,

it is immediately seen that, in the limit ε → 0, the po-

tentially dangerous term f(x)/r(2+2ε)1 in X(V1V1V2) does

not give any contribution to the Hadamard-regular partof X. This means that we can compute the ε-regularizedreduced tidal action in (2.12) by replacing, from the start,the brick potentials V (x), Vi(x), . . ., by their Hadamard-regularized counterparts, R [V (x)], R [Vi(x)], . . .

Summarizing: The A-worldline part of the tidal actionEq. (2.12) can be obtained by computing all its elements(dτA = c−1(−gµν(yA) dyµ

A dyνA)1/2, GA

αβ , . . .) within the

A-regular metric gA−regµν (x) obtained by replacing each

2PN brick potential V (x), Vi(x), . . . by its A-Hadamard-regular part RA[V (x)], RA[Vi(x)], . . .

As a check on our results (and on the many compli-cated algebraic operations needed to derive them) wehave also re-computed the electric-quadrupole tidal La-

grangian, Lµ(2)A

= 14 (dτA/dt) G

Aαβ G

αβA by effecting the

Hadamard regularization in a different way. Our alter-native computation was done by separately Hadamard-regularizing each factor entering the Lagrangian, L

µ(2)A

,

when it is expressed in terms of dτA/dt, the contravari-ant metric, the covariant Riemann tensor, and the con-travariant 4-velocity. More precisely, we first calculatedGαβ(yA) as −RA[Rαµβν ]RA[uµ]RA[uν ], then we com-puted [G2

ab](yA) ≡ Gαβ(yA)Gµν(yA)RA[gαµ]RA[gβν ],which we inserted into the expression of L

µ(2)A

just written. The remaining factor, (dτA/dt)/4,was taken to be RA[dτ/dt]/4. Note in passingthat, while one can a priori prove that the alter-native regularization of Gαβ(yA) (and subsequently[G2

ab](yA) ≡ Gαβ(yA)Gµν(yA)RA[gαµ]RA[gβν]) just ex-plained, must coincide with the one explained above,namely (Gαβ G

αβ)[RA(V ), RA(Vi), . . .] (because both ofthem agree with the Riesz-analytic-regularization and/ordimensional-regularization) a different result would havebeen obtained if one had postponed the Hadamardregularization of the squared tidal quadrupole to thelast moment, i.e. if one had computed RA[Gαβ G

αβ ].[Such a difference occurs because of the appearance ofa dangerous nonlinear mixing of Hadamard-regular andHadamard-singular parts in ∂ijV ∂ijX

(V1V1V2) (with the

special structure of the delicate terms in X(V1V1V2) givenabove). This shows again the consistency problems ofthe Hadamard regularization, when it is used beyondthe types of calculations where it is equivalent to theRiesz analytic regularization (or to dimensional regular-ization).]

D. Explicit rules for computing the regular partsof the 2PN brick potentials

Let us now give some indications on the computa-tion of the regular parts of the various brick potentialsV (x), Vi(x), . . .

1. Regularizing V and Vi

The situation is very simple for the “linear poten-tials” V and Vi, which satisfy linear equations with delta-function sources [see Eqs. (3.3)]. Near, say, the particleA = 1, the A-regular parts of V and Vi are the terms inEqs. (A1), (A2) which are generated by the source terms∝ δ(x−y2) of the second particle. It is indeed easily seen[from the definition in Eq. (3.10)] that the 1-regular partof all the terms explicitly written in Eq. (A1) vanishes,while all the non-explicitly written terms obtained by the1 ↔ 2 exchange are regular near the particle 1. The sameis true for Vi, Eq. (A2). A simple rule for obtaining theseresults is to note that, from the definition in Eq. (3.11),any term of the form

r2k+11 f(x) , k ∈ Z , (3.12)

9

where f(x) is a smooth function of xµ (near x = y1

at fixed instant t), and where the power of r1 is odd, ispurely singular.

The situation is more complicated for the higher-orderpotentials Wij and Ri, whose sources contain both com-pact terms ∝ δ(x−yA), and quadratically nonlinear non-compact ones ∝ ∂V ∂V , and still more complicated forthe X potential whose source even depends on the pre-vious Wij potential.

2. Regularizing Wij

The potential Wij can be decomposed in powers of themasses. It contains terms proportional to m1,m2,m

21,m

22

and m1m2. It is easily seen that while the terms propor-tional to m1 and m2

1 are 1-singular, the terms propor-tional to m2 and m2

2 are 1-regular. It is more delicate todecompose the mixed terms ∝ m1m2 into 1-regular (R1)and 1-singular (S1) parts. More precisely the m1m2 part

of Wij has the form

W[m1m2]ij = W

[m1m2]ij(0) +

˜W

[m1m2]ij(0) (3.13)

where

Wij[m1m2](0) =

1

r12Sδij +

1

S2

(n

(i1 n

j)2 + 2n

(i1 n

j)12

)

− ni12n

j12

(1

S2+

1

r12S

)

≡ 1

r12SP (n12)

ij

+1

S2

(n

(i1 n

j)2 + 2n

(i1 n

j)12 − ni

12nj12

), (3.14)

˜Wij

[m1m2](0) =

1

r12Sδij +

1

S2

(n

(i2 n

j)1 − 2n

(i2 n

j)12

)

− ni12n

j12

(1

S2+

1

r12S

)

≡ 1

r12SP (n12)

ij

+1

S2

(n

(i2 n

j)1 − 2n

(i2 n

j)12 − ni

12nj12

), (3.15)

and where P (n12)ij ≡ δij−ni

12 nj12 denotes the projector

orthogonal to the unit vector n12. [The decompositionin Eq. (3.13) simply corresponds to the decompositionof Eq. (A4) into an explicitly written term and its 1 ↔ 2counterpart.] Here we see that there appear (modulox-independent factors, such as r−1

12 , ni12, P (n12)

ij , . . .)terms of the type

1

S,

1

S2,

ni1

S2,

ni2

S2,

ni1 n

j2

S2, (3.16)

where we recall that S ≡ r1 + r2 + r12. Near particle 1,ni

2 is a smooth function, while ni1 = ri

1/r1 is the ratio of

a smooth function (ri1 = xi − yi

1) by r1. In other words,the five terms listed in Eq. (3.16) are of three differenttypes:

1

S,

f(x)

S2and

f(x)

r1 S2, (3.17)

where f(x) denotes a generic smooth function near parti-cle 1. [As we always consider the neighborhood of particle1, we do not add an index to f(x) to recall that it is 1-regular, but might be singular near particle 2.] BecauseS = r1+r2+r12 is a function of “mixed character” (partlyregular and partly singular), it is not immediate to de-compose the functions in Eq. (3.17) into 1-regular and1-singular parts. [This mixed character of S is deeplylinked with the fact that it enters the 2PN metric be-cause of the basic fact that a solution of ∆g = r−1

1 r−12

is g = ln S.] A simple (though somewhat brute-force)way of extracting the regular parts of the functions inEq. (3.17) consists of decomposing S into

S ≡ S0 + r1 = S0

(1 +

r1S0

), (3.18)

with

S0 ≡ r2 + r12 , (3.19)

(note that S0 is a smooth function near particle 1), andthen expanding S−n in powers of r1/S0. Namely

1

S=

1

S0

(1− r1

S0+r21S2

0

− r31S3

0

+ . . .

), (3.20)

1

S2=

1

S20

(1− 2

r1S0

+ 3r21S2

0

− 4r31S3

0

+ . . .

), (3.21)

and more generally

1

Sn=

1

Sn0

(1− n

r1S0

+(n+ 1)n

2

(r1S0

)2

− (n+ 2)(n+ 1)n

3!

(r1S0

)3

+(n+ 3)(n+ 2)(n+ 1)n

4!

(r1S0

)4

+ . . .

),

n = 1, 2, . . . (3.22)

Using these expansions, together with the rule that termsof the form in Eq. (3.12) are purely singular, it is easy toderive the following results for the 1-regular parts of func-tions of the type in Eq. (3.17), and, more generally, of thetypes f(x)/S, f(x)/S2, f(x)/(r1 S) and f(x)/(r1 S

2):

(f(x)

S

)

R

=f(x)

S0

(1 +

(r1S0

)2

+

(r1S0

)4

+

(r1S0

)6

+ . . .

)

10

≡ f(x)

(1

S

)

R

, (3.23)

(f(x)

S2

)

R

=f(x)

S20

(1 + 3

(r1S0

)2

+ 5

(r1S0

)4

+ 7

(r1S0

)6

+ . . .

)

≡ f(x)

(1

S2

)

R

, (3.24)

(f(x)

r1S

)

R

= −f(x)

S20

(1 +

(r1S0

)2

+

(r1S0

)4

+ . . .

)

≡ f(x)

(1

r1S

)

R

, (3.25)

(f(x)

r1S2

)

R

= −f(x)

S30

(2 + 4

(r1S0

)2

+ 6

(r1S0

)4

+ 8

(r1S0

)6

+ . . .

)

≡ f(x)

(1

r1S2

)

R

. (3.26)

Here, we use a lower R subscript (ϕ(x))R to denotethe 1-regular part of a function ϕ(x) (above denoted asR [ϕ(x)]). (We omit decorating R with a label 1, but oneshould remember that we are always talking about the1-regular part of ϕ(x).)

Note that, as indicated, all the terms above have thesimple property that the regular-projection operator Rcommutes with the multiplication by a smooth function,e.g. R [f(x)S−1] = f(x)R [S−1]. Beware that this prop-erty is true only for the special singular terms consideredhere. We shall later see that more-complicated singu-lar terms (entering the X potential) do not satisfy thissimple commutativity property.

Note that the number of terms one needs to retain inthe above expansions depends on the quantity one wantsto evaluate on the first worldline. For instance, whenevaluating G1

αβ , which involves the curvature tensor, and

therefore two spatial derivatives of the metric (and, in

particular, of R [Wij ]), we need to include enough terms

to ensure that R [Wij ] is C2 near x = y1. Actually, weshall push our calculations up to the level of G1

αβγ , whichdepends on the first covariant derivative of the curvaturetensor, and we shall therefore need all the brick potentialsto be at least C3 near x = y1.

The application of the above results yields the follow-ing explicit expressions for the 1-regular part of the twoseparate O(m1m2) delicate contributions to Wij [definedin Eqs. (3.13)–(3.15)]:

[Wij[m1m2](0) ]R =

P (n12)ij

r12

1

S0

(1 +

r21S2

0

+r41S4

0

+ . . .

)

+r(i1 n

j)2

S20

(− 2

S0− 4

r21S3

0

− 6r41S5

0

+ . . .

)

+ 2r(i1 n

j)12

S20

(− 2

S0− 4

r21S3

0

− 6r41S5

0

+ . . .

)

− ni12n

j12

1

S20

(1 + 3

r21S2

0

+ 5r41S4

0

+ . . .

),

(3.27)

[˜Wij

[m1m2](0) ]R =

P (n12)ij

r12

1

S0

(1 +

r21S2

0

+r41S4

0

+ . . .

)

+n

(i2 r

j)1

S20

(− 2

S0− 4

r21S3

0

− 6r41S5

0

+ . . .

)

− 2n(i2 n

j)12

1

S20

(1 + 3

r21S2

0

+ 5r41S4

0

+ . . .

)

− ni12n

j12

1

S20

(1 + 3

r21S2

0

+ 5r41S4

0

+ . . .

).

(3.28)

3. Regularizing Ri

As the potential Ri has a source of the same type asWij (namely δ(x − yA) terms plus a non-compact termquadratic in the V potentials), the calculation of its reg-

ular part can be done in exactly the same way as Wij . Ri

contains terms ∝ m21,m

22 and m1m2. The O(m2

1) pieceis purely singular, the O(m2

2) one is purely regular, whilethe O(m1m2) one is a mix of regular and singular terms.

As above, we can decompose the m1m2 part of Ri in twopieces, say

R[m1m2]i = R

[m1m2]i(0) +

˜R

[m1m2]i(0) , (3.29)

where

Ri[m1m2](0) = ni

12

− (n12v1)

2S

(1

S+

1

r12

)

− 2(n2v1)

S2+

3(n2v2)

2S2

+ ni1

1

S2

(2(n12v1)−

3(n12v2)

2

+ 2(n2v1)−3(n2v2)

2

)

+ vi1

(1

r1r12+

1

2r12S

)− vi

2

1

r1r12, (3.30)

˜Ri

[m1m2](0) = −ni

12

(n12v2)

2S

(1

S+

1

r12

)

− 2(n1v2)

S2+

3(n1v1)

2S2

11

+ ni2

1

S2

(−2(n12v2) +

3(n12v1)

2

+ 2(n1v2)−3(n1v1)

2

)

+ vi2

(1

r2r12+

1

2r12S

)− vi

1

1

r2r12. (3.31)

Applying the above results then yields the followingexpressions for the 1-regular parts of these quantities:

[Ri[m1m2](0) ]R = ni

12

[− (n12v1)

2− 2(n2v1)

+3(n2v2)

2

](1

S2

)

R

− (n12v1)

2r12

(1

S

)

R

+

(2(n12v1)−

3(n12v2)

2+ 2(n2v1)

−3(n2v2)

2

)r

i1

(1

r1S2

)

R

+vi1

2r12

(1

S

)

R

, (3.32)

[˜Ri

[m1m2](0) ]R = −ni

12

(n12v2)

2

((1

S2

)

R

+1

r12

(1

S

)

R

)− 2(r1v2)

(1

r1S2

)

R

+3

2(r1v1)

(1

r1S2

)

R

+ ni2

[(−2(n12v2) +

3(n12v1)

2

)(1

S2

)

R

+

(2(r1v2)−

3(r1v1)

2

)(1

r1S2

)

R

]

+ vi2

(1

r2r12+

1

2r12

(1

S

)

R

)− vi

1

1

r2r12.

(3.33)

One should substitute the expansions in Eqs. (3.23)–(3.26) into these results to get their explicit forms.

4. Regularizing X

Finally, we come to the most complicated 2PN brickpotential, namely X . It contains contributions propor-tional to m2

1,m1m2,m22; m

31,m

21m2,m1m

22 and m3

2 (seeEq. (A5)). The terms in m2

1,m22,m

31,m

32 are easily dealt

with (they are either purely singular or purely regular).Many, but not all, of the m1m2 terms can be dealt within the same way as the m1m2 terms in Wij and Ri. If

we again decompose X [m1m2] in two pieces

X [m1m2] = X[m1m2](0) +

˜X

[m1m2](0) , (3.34)

we have the following results for their regular parts:

[X[m1m2](0) ]R = v2

1

[(1

r1S

)

R

+1

r12

(1

S

)

R

]+ v2

2

[(1

r1S

)

R

+1

r12

(1

S

)

R

]− (v1v2)

(2

(1

r1S

)

R

+3

2r12

(1

S

)

R

)

−(n12v1)2

((1

S2

)

R

+1

r12

(1

S

)

R

)− (n12v2)

2

((1

S2

)

R

+1

r12

(1

S

)

R

)

+3(n12v1)(n12v2)

2

((1

S2

)

R

+1

r12

(1

S

)

R

)+ 2(n12v1)(r1v1)

(1

r1S2

)

R

−5(n12v2)(r1v1)

(1

r1S2

)

R

− (r1v1)2

(1

r21S2

+1

r31S

)

R

+ 2(n12v2)(r1v2)

(1

r1S2

)

R

+2(r1v1)(r1v2)

(1

r21S2

+1

r31S

)

R

− (r1v2)2

(1

r21S2

+1

r31S

)

R

− 2(n12v2)(n2v1)

(1

S2

)

R

+2(r1v2)(n2v1)

(1

r1S2

)

R

− 3

2(r1v1)(n2v2)

(1

r1S2

)

R

, (3.35)

[˜X

[m1m2](0) ]R = v2

2

(1

r2r12+

1

r2

(1

S

)

R

+1

r12

(1

S

)

R

)+ v2

1

(− 1

r2r12+

1

r2

(1

S

)

R

+1

r12

(1

S

)

R

)

−(v1v2)

(2

r2+

3

2r12

)(1

S

)

R

− (n12v2)2

((1

S2

)

R

+1

r12

(1

S

)

R

)

−(n12v1)2

((1

S2

)

R

+1

r12

(1

S

)

R

)+

3(n12v2)(n12v1)

2

((1

S2

)

R

+1

r12

(1

S

)

R

)

12

−2(n12v2)(n2v2)

(1

S2

)

R

+ 5(n12v1)(n2v2)

(1

S2

)

R

−(n2v2)2

((1

S2

)

R

+1

r2

(1

S

)

R

)− 2(n12v1)(n2v1)

(1

S2

)

R

+2(n2v2)(n2v1)

((1

S2

)

R

+1

r2

(1

S

)

R

)− (n2v1)

2

((1

S2

)

R

+1

r2

(1

S

)

R

)

+2(n12v1)(r1v2)

(1

r1S2

)

R

+ 2(n2v1)(r1v2)

(1

r1S2

)

R

− 3

2(n2v2)(r1v1)

(1

r1S2

)

R

. (3.36)

One should substitute the expansions in Eqs. (3.23)–(3.26) into the corresponding terms in Eqs. (3.35), (3.36).However, these equations involve new types of terms, notdiscussed above. These new terms are of the form

f(x)

(1

r21 S2

+1

r31 S

). (3.37)

The fact that we have a specific combination of r−21 S−2

and r−31 S−1 simplifies things. Indeed, using the expan-

sions in Eqs. (3.20), (3.21) above we have

f(x)

r21 S2

=f(x)

S40

[S2

0

r21− 2

S0

r1+ 3− 4

r1S0

+ 5r21S2

0

+ . . .

],

(3.38)

f(x)

r31 S=f(x)

S40

[S3

0

r31− S2

0

r21+S0

r1− 1 +

r1S0− r21S2

0

+ . . .

].

(3.39)When summing these two equations we see that the terms∝ 1/r21 cancel. We shall deal later with these terms,which turn out to be delicate to handle but, anyway, inthe sum of Eqs. (3.38) and (3.39), they cancel out. Theremaining terms contain either an odd power of r1 [andare therefore purely singular, Eq. (3.12)] or a positive,even power of r1 (which makes them purely regular). Asa consequence, the regular part of the combination of Eq.(3.37) reads

[f(x)

(1

r21S2

+1

r31S

)]

R

=f(x)

S40

(2 + 4

(r1S0

)2

+6

(r1S0

)4

+ 8

(r1S0

)6

+ . . .

)

≡ f(x)

(1

r21S2

+1

r31S

)

R

.

(3.40)

Note that, thanks to the cancellation of the 1/r21terms, we have again a property of commutativityR [f(x)ϕ(x)] = f(x)R [ϕ(x)], for the special type ofterms ϕ(x) entering Eq. (3.37).

Concerning the m1m22 contribution to X, it is the sum

of

X[m1m2

2]

(0) = − 1

2r312+

r22r1r312

− 1

2r1r212(3.41)

and

˜X

[m1m22]

(0) =1

2r32+

1

16r31+

1

16r22r1− r21

2r22r312

+r31

2r32r312

− r2232r31r

212

− 3

16r1r212+

15r132r22r

212

− r212r32r

212

− r12r32r12

− r21232r22r

31

. (3.42)

Using the rule of Eq. (3.12), we easily see that eachterm clearly is either purely regular or purely singular.Computing the regular part of X [m1m2] is then easy.

The most delicate contribution to X is its O(m21m2)

one, which can again be written as the sum of

X[m2

1m2]

(0) =1

2r31+

1

16r32+

1

16r21r2− r22

2r21r312

+r32

2r31r312

− r2132r32r

212

− 3

16r2r212+

15r232r21r

212

− r222r31r

212

− r22r31r12

− r21232r21r

32

(3.43)

and

˜X

[m21m2]

(0) = − 1

2r312+

r12r2r312

− 1

2r2r212. (3.44)

Actually the part˜X

[m21m2]

(0) is easy to discuss: Its reg-

ular part is obtained simply by discarding the term:

r1/(2 r2 r312). Similarly, most of the terms in X

[m21m2]

(0)

are easy to treat, being either purely regular or purelysingular because of Eq. (3.12). However, the third,fourth, eighth and last terms in the right hand side ofEq. (3.43) are somewhat tricky. [These terms correspond

to the “dangerous terms” in X(V1V1V2) that were dis-cussed in Sec. III when making the link between theε-regularization and the Hadamard one.] The third termis

Q ≡ 1

16 r21 r2, (3.45)

while the sum of the fourth, eighth and last terms reads

P ≡ − r222 r21 r

312

+15 r2

32 r21 r212

− r21232 r21 r

32

. (3.46)

13

Both Q and P are of the form f(x)/r21 (but we shall seethat Q is special compared with P ). The computationof the regular part of f(x)/r21 is a bit subtle. It can,however, be done by brute force, namely by replacing thesmooth function f(x) by its Taylor expansion around y1:

f(x) = f(y1) + ri1 ∂i f(y1) +

1

2!ri1 r

j1 ∂ij f(y1)

+1

3!ri1 r

j1 r

k1 ∂ijk f(y1) + . . . (3.47)

When replacing ri1 → r1 n

i1 and dividing by r21 , one sees

that the regular part of f(x)/r21 will only come from the

terms rL1 ≡ ri1i2...iℓ

1 with ℓ = 2, 4, 6, . . . Moreover, bydecomposing rL

1 = rℓ1 n

L1 in irreducible tensorial parts, as

in

rij1 = r21 n

ij1 = r21

[n〈ij〉1 +

1

3δij

], (3.48)

where n〈ij〉1 ≡ nij

1 ≡ nij1 − 1

3 δij denotes the symmetric

trace-free projection of nij1 ≡ ni

1 nj1, we see [in view of the

definition in Eq. (3.10)] that only the pieces containingat least one Kroneker δ in the decomposition of nL

1 willcontribute to the regular part. For instance, in the caseℓ = 2, only the δij in Eq. (3.47) will contribute to theregular part of f(x)/r21 . More generally, we have thatR[rL

1 /r21 ] = (rL

1 − rL1 )/r21 .

Applying this method yields the following result (herewritten with the simplified notation used around Eq.(3.8)) for the regular part of f(x)/r21 :

(f(x)

r2

)

R

=1

6∆f(0) +

1

10xi∂i∆f(0) +

1

28xij∂ij∆f(0)

+1

120r2∆2f(0) +

1

108xijk∂ijk∆f(0)

+1

280r2xi∂i∆

2f(0) +O(x4) . (3.49)

As one sees in Eq. (3.49) (and as can be proven toall orders), all the terms on the right-hand side of Eq.(3.49) are derivatives of the Laplacian of f(x) (taken atx = y1). As a consequence, in the particular case where∆f(x) = 0, the regular part of f(x)/r21 is exactly zero.

This is the case for the term Q in X [m21m2], Eq. (3.45).

[Let us point out in passing that the discussion in Sec.IIIC of the link between the ε-regularization and theHadamard one essentially consisted in remarking that all

the “dangerous” terms in X [m21m2] had this innocuous

structure f(x)/r(2+2ε)1 with ∆f(x) = 0.] Therefore, we

have simply

QR = 0 . (3.50)

On the other hand, this is not the case for the termP , Eq. (3.46). The evaluation of the regular part of Pneeds to appeal to the result in Eq. (3.49) and yields

(modulo terms of order O(r41) that will not be needed in

our calculations)

PR = − 1

2r312

(r22r21

)

R

+15

32r212

(r2r21

)

R

− r21232

(1

r21r32

)

R

= − 3

8r312+

1

r512

[3

224r21 −

15

112(r1n12)

2

]

+5

12r612(r1n12)

[(r1n12)

2 − 3

8r21

]. (3.51)

Summarizing: We have explicitly displayed all therules needed to compute (near particle 1) the regular

parts of the various brick potentials V, Vi, Wij , Ri, X en-

tering the 2PN metric. By replacing V → VR, . . . , X →XR, in Eq. (3.2), we define a regularized version of the2PN metric generated by two point masses, gR

µν(x) ≡gµν [VR(x), . . . , XR(x)], which is smooth near particle 1.

IV. COMPUTATION OF THE INVARIANTSENTERING THE TIDAL ACTION

As we explained above, when neglecting termsquadratic in the tidal parameters µ(ℓ), etc., the tidal partof the two-body action is simply obtained by evaluatingthe Snonminimal, Eq. (2.12), as a function of the world-lines, by replacing the metric gµν(x) entering the right-hand side of Eq. (2.12) by the (regular part of the) point-mass metric gpointmass

µν (x, y1, y2,m1,m2). This reduced

action is a sum over the various tidal parameters, µ(ℓ)A ,

σ(ℓ)A , µ

′(ℓ)A , . . .. We can therefore compute separately the

part of the reduced action associated with each of them.This is what we shall do in this section for the actionsassociated with the parameters µ

(ℓ=2)A=1 , µ

(ℓ=3)A=1 , σ

(ℓ=2)A=1 and

µ′(ℓ=2)A=1 . [We shall only explicitly write down the results

for A = 1 but, evidently, they also yield the results forA = 2 by exchanging 1 ↔ 2.]

First, let us note that each action, say, associated withthe parameter µ1 related to the first worldline, is of theform

µ1

∫dt Lµ1(y1,y2,v1,v2) (4.1)

where the Lagrangian Lµ1 is the product of a geometricalinvariant by dτ1/dt. For instance

Lµ(2)1

=1

4

dτ1dt

[Gαβ G

αβ]1≡ 1

4

dτ1dt

[G2

ab

]1. (4.2)

We shall separately evaluate each geometrical invari-ant, G2

ab, G2abc, . . ., before multiplying it by the (regular-

ized) proper-time redshift factor dτ1/dt (“Einstein timedilation”). Note also that we systematically work withthe order-reduced 2PN metric, i.e. the 2PN metric inwhich the higher time derivatives of y1 and y2 have

14

been expressed in terms of positions and velocities only,(y1,y2,v1,v2), by iterative use of the (harmonic-gauge)equations of motion. As was discussed long ago, such anorder reduction of the action is allowed, when it is under-stood that it corresponds to a certain additional changeof coordinate gauge [45–47]. As we shall ultimately beinterested in computing gauge-invariant quantities asso-ciated with the EOB reformulation of the dynamics, wedo not need to keep track of this coordinate change.

A. Explicit 2PN-accurate tidal actions for generalorbits

Let us start by discussing the simplest (and physicallymost important) geometric invariant, namely the one as-sociated with the electric-type quadrupolar tide, say

J2e ≡ [GabGab]1 = [Gαβ Gαβ ]1

= [Rαµβν Rγκδλ gαγ gβδ uµ uν uκ uλ]1 , (4.3)

where uµ1 ≡ dyµ

1 /dτ1 = (c,v1) dt/dτ1, and where the sub-script 2e on J2e refers to “ℓ = 2 electric.” Using twoindependently written codes (one based on the Maplesystem, and the other one based on the Mathematicasoftware supplemented by the package xAct [48]) wehave computed the right-hand side of Eq. (4.3) withinthe (regularized) 2PN metric. (Actually, as explainedabove, the Mathematica code alternatively regularized,a la Hadamard, the value of Gαβ computed with the full(non-regularized) 2PN metric.)

As the PN expansion of the quadrupolar tidal tensorEq. (2.7) starts as

Gab = − c2Ra0b0 + . . .

= +1

2c2(∂ab g00 − ∂a0 gb0 − ∂b0 ga0 + ∂00 gab)

+ . . . ,

one sees that the 2PN-accurate metric [i.e., the knowl-edge of g00 up to O(1/c6) terms included, of g0a up toO(1/c5), and of gab up to O(1/c4)] is exactly what isneeded to be able to compute Gab to the 2PN (frac-tional) accuracy, i.e., Gab = + ∂abV +c−2(. . .)+c−4(. . .).The same is true for the higher electric tidal momentsGabc, . . .. However, one can easily see that one losesa PN order when evaluating either the magnetic tidalmoments Hab, Habc, . . . or the time-differentiated electricone Gab, . . .. The result we obtained, for general orbits,is

J2e =6G2m2

2

r612

1 +

1

c2

(− 3(n12v12)

2 − 3(n12v2)2 + 3v2

12

− G

r12(5m1 + 6m2)

)

+1

c4

[3(n12v12)

4 + 12(n12v2)2(n12v12)

2 + 6(n12v2)4

−9v212(n12v12)

2 − 6(n12v12)2(v2v12)

−6(n12v2)(n12v12)(v2v12)− 3v22(n12v12)

2

−9v212(n12v2)

2 − 3v22(n12v2)

2 + 6v412

+6v212(v2v12) + 3(v2v12)

2 + 3v22v

212

+Gm1

r12

(− 109

4(n12v12)

2 +41

2(n12v2)

2 +21

4v212

)

+Gm2

r12

(6(n12v12)

2 + 21(n12v2)2 − 6v2

12

)

+G2

r212

(365m21

28+

125m1m2

2+ 21m2

2

)]. (4.4)

Similarly, we computed the further geometrical invariants“ℓ = 3 electric”

J3e ≡[G2

abc

]1

=[Gαβγ G

αβγ]1, (4.5)

and “ℓ = 2 magnetic”

J2m ≡ 1

4

[H2

ab

]1

=1

4

[Hαβ H

αβ]1

= c2[R∗αµβν R

∗γκδλ g

αγ gβδ uµ uν uκ uλ]1.(4.6)

Note the factor 14 , introduced in the definition of

J2m to have J2m = (cR∗αµβνuµuν)2, analogously to

J2e = (Rαµβνuµuν)2. Let us also note in passing that,

in evaluating J3e, i.e., the square of the electric oc-tupole Gαβγ , Eq. (2.8), it is important to use the or-thogonally projected covariant derivative ∇⊥

α . If, in-stead of (Gαβγ)2, one evaluates (Cαβγ)2 where Cαβγ =Symαβγ ∇αRβµγν u

µ uν , one finds a result which differs

from (Gαβγ)2 by a term proportional to J2e = (Gab)2

[see Eq. (6.51)].The results for these invariants (along general orbits)

are

J3e =90G2m2

2

r812

1 +

1

c2

[− 2(n12v12)

2 − 4(n12v2)2 + 3v2

12

−G(4m1 + 10m2)

r12

]

+1

c4

[10(n12v2)

2(n12v12)2 + 10(n12v2)

4

−14

3v212(n12v12)

2 − 4(n12v12)2(v2v12)

−12v212(n12v2)

2 − 4(n12v2)(n12v12)(v2v12)

−2v22(n12v12)

2 − 4v22(n12v2)

2

+17

3v412 + 6v2

12(v2v12) + 3(v2v12)2 + 3v2

2v212

+Gm1

r12

(− 32(n12v12)

2 + 2(n12v2)(n12v12)

+22(n12v2)2 +

16

3v212

)

+Gm2

r12

(12(n12v12)

2 + 45(n12v2)2 − 18v2

12

)

+G2

r212

(9m2

1 +259m1m2

3+ 54m2

2

)], (4.7)

15

and

J2m =18G2m2

2

r612

− (n12v12)

2 + v212

+1

c2

[(n12v12)

4 + 4(n12v2)2(n12v12)

2

−3v212(n12v12)

2 − 2(n12v12)2(v2v12)− v2

2(n12v12)2

−2(n12v2)(n12v12)(v2v12)− 3v212(n12v2)

2

+2v412 + 2v2

12(v2v12) + (v2v12)2 + v2

2v212

+2G

r12

(m1

3+m2

)((n12v12)

2 − v212

)]. (4.8)

The result Eq. (4.4), after multiplication by the red-shift factor

dτ1dt

=

(− g00 − 2 g0i

vi1

c− gij

vi1 v

j1

c2

)1/2

, (4.9)

which evaluates to (we use again the notation ǫ ≡ 1/c,and henceforth often set Newton’s constant to one)

dτ1dt

= 1− ǫ2(

1

2v21 +

m2

r12

)

+ ǫ4[−1

8v41 +

m2

r12

(1

2(n12v2)

2 − 3

2v21

+ 4(v1v2)− 2v22

)+

m2

2r212(3m1 +m2)

], (4.10)

provides the O(µ(ℓ=2)1 ) piece (“gravitoelectric tidal

quadrupole”) of the reduced two-body action at the 2PNapproximation level; i.e., including tidal correction termsthat are (v/c)4 smaller than the leading order tidal La-

grangian which is simply given by J(0)2e = 6m2

2/r612. Sim-

ilarly, multiplying the results of Eqs. (4.7) and (4.8)by the redshift factor in Eq. (4.10) provides the re-duced tidal actions associated with J3e ≡ [G2

abc]1 andJ2m ≡ 1

4 [H2ab]1, at the 2PN approximation for the

electric-octupole term J3e, and at the 1PN approxima-tion for the magnetic-quadrupole term J2m.

In view of their complexity, the results of Eqs. (4.4),(4.7), (4.8), which provide the action for general orbits,are not very useful as they are. In what follows, we shallextract the physically most useful information they con-tain by: (i) focusing our attention on circular orbits and(ii) reformulating our results in terms of the EOB de-scription of binary systems. Note in passing that thoughcircular orbits are only special solutions of binary dynam-ics, they are the ones of prime physical importance inmany situations, most notably radiation-reaction-driveninspiralling binary systems.

B. Tidal actions along circular orbits

In the following, we shall therefore restrict our atten-tion to circular motions. [However, we shall show below

how this restricted result can crucially inform the EOBdescription of tidally interacting binary systems.] Weshall also focus on the relative dynamics in the center ofmass frame. As we see in Eqs. (4.4), (4.7), (4.8), (4.10),the various Lagrangians depend only on the relative po-sition y12 = y1−y2 and start depending on (individual)velocities only at 1PN (for general orbits), and even at2PN for the invariants themselves (in the case of circu-lar orbits). This implies that we shall not really need touse to its full 2PN accuracy the relation between center-of-mass variables yCM

1 ,yCM2 ,vCM

1 ,vCM2 , and relative ones

y12,v12, namely (in the circular case)

yi1 =

[X2 + 3

(M

r12 c2

)2

ν X12

]yi12 ,

yi2 =

[−X1 + 3

(M

r12 c2

)2

ν X12

]yi12 , (4.11)

and the corresponding velocity relations obtained bytime-differentiating them, using the fact that y12 =r12 n12 where r12 is constant and n12 rotates with anangular velocity given by

Ω2 =M

r312

[1 + ǫ2(ν − 3)

M

r12

+ǫ4(

6 +41

4ν + ν2

)(M

r12

)2]. (4.12)

Here and below we use the notation

X1 ≡m1

M, X2 ≡

m2

M, ν ≡ X1X2 , X12 ≡ X1−X2 ,

(4.13)(recall that M ≡ m1 +m2 so that X1 +X2 = 1). In ourcalculations, the ǫ4 = 1/c4 contributions in Eqs. (4.11),(4.12) do not matter, and can be neglected from the start.

Using such an additional circular (and center-of-mass)reduction, we get a much simplified result for the electric-quadrupole invariant J2e, Eq. (4.3), namely,

J(circ)2e =

6M2X22

r612

[1 + ǫ2

(X1 − 3)M

r12

−ǫ4 M2

28r212(713X2

1 − 805X1 − 336)

]. (4.14)

In a similar manner, one gets much simplified results forthe other (subleading) geometrical invariants of tidal sig-nificance, namely the magnetic quadrupolar term J2m,Eq. (4.6), the electric octupolar term J3e, Eq. (4.5), andalso for the time-differentiated electric-quadrupole cou-pling, say,

J2e ≡[G2

ab

]1

=[(uµ∇µ Gαβ)(uν ∇ν G

αβ)]1. (4.15)

Among these invariants, the 2PN accurate metric allowsone (as for G2

ab) to calculate to 2PN fractional accuracy

16

only the electric-octupole term J3e. The other ones canbe computed only at 1PN fractional accuracy becauseof their “magnetic,” or “∂0 = c−1∂t” character. Ourexplicit “circular” results for J2m, J3e and J2e are

J(circ)2m =

18X22M

3

r712

[1 + ǫ2

M

3r12(3X2

1 +X1 − 9)

],

(4.16)

J(circ)3e =

90X22M

2

r812

[1 + ǫ2(6X1 − 7)

M

r12

−ǫ4 M2

3r212(61X2

1 + 4X1 − 98)

], (4.17)

J(circ)

2e=

18X22M

3

r912

[1 + ǫ2(X2

1 − 7)M

r12

]. (4.18)

To complete the above results, and allow one to com-pute the corresponding associated Lagrangians, let usnote that the circular value of the redshift factor is

dτ1dt

≡ 1

Γ1= 1− M(X1 − 1)(X1 − 3)

2r12ǫ2

+M2(X1 − 1)

8r212(3X3

1 − 9X21 + 13X1 − 3)ǫ4 . (4.19)

Let us also quote the value of the inverse redshift factor,Γ1 (analog to a Lorentz γ-factor γ = 1/

√1− v2/c2),

namely

Γ1 ≡dt

dτ1= 1 +

M(X1 − 1)(X1 − 3)

2r12ǫ2

− M2

8r212(X1 − 1)(X3

1 + 5X21 − 17X1 + 15)ǫ4 . (4.20)

V. EOB DESCRIPTION OF THE TIDALACTION

We have computed above the effective actions associ-

ated with the tidal parameters µ(2)1 , µ

(3)1 , σ

(2)1 and µ

′(2)1 .

Before the restriction to circular motions (in the center-of-mass frame) they have the general form

µ1

∫dt Lµ1(y12,v1,v2) , (5.1)

where µ1 denotes a generic tidal parameter, and y12 =y1 − y2. In this section we discuss how one can de-scribe the actions of Eq. (5.1) within the EOB formal-ism. Let us recall that the EOB formalism [16–19] re-places the (possibly higher-order) Lagrangian dynamicsof two particles by the Hamiltonian dynamics of an “effec-tive particle” embedded within some “effective externalpotentials.” For non-spinning [66] bodies, the original

(velocity-dependent) two-body interactions become re-formulated (and simplified by means of a suitable contacttransformation in phase space) in terms of three “EOBpotentials”: A(reff), B(reff) and Q(reff , p

eff). The firsttwo potentials, A(reff) and B(reff), parametrize an “ef-fective metric”

ds2eff = gµν(xeff) dxµeff dx

νeff

= −A(reff) c2 dt2eff + B(reff) dr2eff+r2eff(dθ2eff + sin2 θeff dϕ

2eff) , (5.2)

and its associated Hamilton-Jacobi equation, while thethird potential Q(reff , p

eff) (which necessarily appearsat 3PN [18]), describes additional contributions to the(Hamilton-Jacobi) mass-shell condition,

0 = µ2 + gµνeff (xeff) peff

µ peffν +Q(reff , p

eff) (5.3)

[where µ ≡ m1m2/M ≡ νM is the reduced mass of thebinary system], that are higher than quadratic in the ef-fective momentum peff . Following the EOB-simplifyingphilosophy of Ref. [18], we shall assume that the third po-tential has been reduced (by a suitable canonical trans-formation) to a form where it vanishes with the radialmomentum peff

r .

In addition, EOB theory introduces a dictionary be-tween the original dynamical variables (positions, mo-menta, angular momentum, energy) and the effectiveones. A crucial entry of this dictionary is a non-trivialtransformation between the original “real” energy, i.e.,the value of the original (total) Hamiltonian H , and the“effective” energy −peff

0 ≡ Heff entering the mass-shellcondition of Eq. (5.3). Because of this transformation,the final EOB-form of the (original, real) Hamiltonianreads (here we set c = 1 for simplicity)

HEOB(xeff ,peff) = M

√1 + 2 ν

(Heff

µ− 1

), (5.4)

where Heff = Heff(xeff ,peff) is given by

Heff =

√A(reff)

(µ2 +

J2eff

r2eff+

(peffr )2

B(reff)+Q(reff , peff)

).

(5.5)Here Jeff ≡ xeff×peff denotes the effective orbital angularmomentum, which, by the EOB dictionary, is actuallyidentified with the original, total (center of mass) orbitalangular momentum J of the binary system: Jeff ≡ J .

A. EOB reformulation of tidal actions: generalorbits

Let us now discuss what the various possible meth-ods are for reformulating an original action of thetype L0(y12,v1,v2, . . .) + µ1 Lµ1(y12,v1,v2) [where µ1

17

stands for a sum over a collection of tidal pa-

rameters µ(2)1 , µ

(2)2 , µ

(3)1 , µ

(3)2 , . . .] into corresponding µ1-

deformations of EOB potentials: A0(reff) + µ1Aµ1 (reff),B0(reff) + µ1 Bµ1(reff), Q0(reff , p

eff) + µ1Qµ1(reff , peff).

The main difficulty in finding the perturbed EOB po-tentials Aµ1 , Bµ1 , and Qµ1 that encode the dynamics ofLµ1 is that such a dynamical equivalence is obtained onlyafter some a priori unknown phase-space contact trans-formation between the EOB phase-space coordinates, sayξeff = (xeff ,peff), and the original (harmonic-coordinate-related) ones, say ξh = (y12,v12). For simplicity, weassume that we have already performed the reduction ofthe original harmonic-coordinate dynamics to its center-of-mass version, in which one can express v1 and v2

in terms of the relative velocity v12 ≡ v1 − v2 and ofy12 ≡ y1 − y2. On the other hand, we do not imme-diately assume that the original Lagrangian dynamics isexpressed in Hamiltonian form. (Let us recall that, aswas found long ago [24, 49], starting at the 2PN level,the harmonic-coordinate dynamics does not admit anordinary Lagrangian, L(y, y), but only a higher-orderone, L(y, y, y). In order to express the 2PN dynamicsin Hamiltonian form, one already needs some (higher-order) contact transformation. However, this transfor-mation is well-known, e.g., Ref. [46], and we do not needto complicate our discussion by explicitly mentionning it.Nonetheless, it will be taken into account in our calcula-tions below.)

The transformation T between ξeff and ξh will have thegeneral structure

ξh = T0(ξeff) + µ1 Tµ1(ξeff) . (5.6)

The unperturbed part T0(ξeff) is known from the pre-vious EOB work [16, 18], but the O(µ1) perturbed partTµ1(ξeff) is unknown, and, actually, is part of the prob-lem which must be solved for reformulating the (per-turbed) harmonic-coordinate dynamics in EOB form.This means, in particular, that it would not be correct totry to compute Aµ1 , Bµ1 and Qµ1 , simply by replacingin the tidal action in Eq. (5.1) the harmonic variables ξhby their unperturbed expression T0(ξeff) in terms of theeffective variables ξeff .

For the general case of non-circular orbits, a universal,correct method for transforming the original LagrangianL(ξh) = L0(ξh) + µ1 Lµ1(ξh) in EOB form consists (asexplained in Ref. [16]) of the following steps: (i) to trans-form the original Lagrangian L(ξh) in Hamiltonian formH(ξH) = H0(ξH) + µ1Hµ1(ξH), where ξH = (q, p) arecanonical coordinates; (ii) to extract the gauge-invariantcontent of H(ξH) by expressing it in terms of action vari-

ables Ia = 12π

∮pa dqa, which yields the Delaunay Hamil-

tonian H(I) = H0(I) + µ1Hµ1(I); (iii) to do the samething for the EOB Hamiltonian, i.e. to compute, as afunctional of the unknown EOB potentials, its Delaunayform HEOB(I) = HEOB

0 (I) + µ1HEOBµ1

(I); and finally(iv) to identify the known H(I) to HEOB(I), which de-pends on the unknown functions Aµ1 , Bµ1 , Qµ1 . This

last step yields (functional) equations for Aµ1 , Bµ1 , Qµ1

and thereby allows one to determine them. [In prac-tice, the functional dependence on A, B,Q is replacedby a much simpler parameter-dependence by using themethod of undetermined coefficients for parametrizinggeneral forms of A, B,Q.] An alternative (and equallyuniversal) method for transforming L(ξh) in EOB form(as used in Ref. [18]) is to add the transformationξh = T (ξeff) to the list of unknowns (using the methodof undetermined coefficients), and to directly solve theset of constraints for T,A, B and Q coming from the re-quirement that HEOB(ξeff , A, B, Q) = H(T (ξeff)). [Onemust then take into account that T is constrained to bea canonical transformation.]

B. EOB reformulation of tidal actions: circularorbits

The 2PN-accurate results, given for several tidal in-teractions in the case of general orbits, in the previoussection, can, in principle, be transformed within the EOBformat by using any of the two methods we just ex-plained. However, from the point of view of current astro-physical applications, one is mainly interested in knowingthe EOB description of (quasi)-circular motions. In thiscase, we know a priori that it is only the A radial po-tential which matters. Knowing this, the question ariseshow to compute the tidal perturbation Aµ1 of the EOB Apotential in the most efficient manner, possibly withouthaving to go through the rather involved, general univer-sal methods recalled above. Fortunately, it is possible todo so by using the following facts.

The first useful fact concerns the relation between thetidal perturbation (in harmonic coordinates) of the La-grangian of the binary system, say

δLh(yh, vh) = µ1 Lµ1(y12,v1,v2) , (5.7)

and the corresponding tidal perturbation (in harmonic-related phase-space coordinates) of the Hamiltonian, say

δHh(yh, ph) ≡ Hhfull(yh, ph)−Hh

tidal-free(yh, ph) . (5.8)

[Strictly speaking, as we recalled above, the harmonic-related (q, p) = (yh, ph) phase-space coordinates involvea supplementary O(1/c4) gauge transformation linked tothe order reduction of L2PN(y, y, y) into Lred

2PN(y′, y′).]Note that here and in the following the notation δQ(ξ)will always refer to the tidal contribution to some func-tion of specified variables, i.e. δQ(ξ) ≡ Qfull(ξ) −Qtidal-free(ξ). One has to be careful about which variablesare fixed as, for instance, the transformation between La-grangian (q, q) and Hamiltonian (q, p) coordinates doescontain a tidal contribution [because δtidalL(yh, yh) doesdepend on velocities]. This being made clear, we havethe well-known universal result about first-order defor-mations of Lagrangians by small parameters, L(q, q) =

18

L0(q, q) + µ1 Lµ1(q, q), namely

δHh(yh, ph) = − δLh(yh, vh) (5.9)

which follows from the properties of the Legendre trans-form.

Let us now apply the second method recalledabove for transforming the “harmonic” HamiltonianHh

full(yh, ph) = Hh0 (yh, ph) + δHh(yh, ph) (where the in-

dex 0 refers to the unperturbed, tidal-free dynamics)into its corresponding EOB formHEOB

full (xEOB, pEOB), de-fined in Eqs. (5.4), (5.5) above. [For clarity, we denotehere the effective-one-body phase-space coordinates byxEOB, pEOB, instead of xeff , peff as above.] The crucialpoint is that the EOB potentials entering the definitionof HEOB

full must be the full, tidally-completed values ofA, B and Q, e.g.

Afull(rEOB) = A0(rEOB) + µ1Aµ1(rEOB)

≡ A0(rEOB) + δA(rEOB) . (5.10)

In other words δHEOB(xEOB, pEOB) ≡HEOB

full (xEOB, pEOB) − HEOB0 (xEOB, pEOB) is ob-

tained by varying the functions A, B andQ (i.e. A(rEOB) = A0(rEOB) + δA(rEOB),etc.) in the definition in Eqs. (5.4), (5.5) ofHEOB

full [xEOB, pEOB;A(rEOB), B(rEOB), Q(rEOB, pEOB)].

This second method for mapping Hhfull(ξh)

into HEOBfull (ξEOB) [where ξh ≡ (yh, ph), ξEOB ≡

(xEOB, pEOB)] consists of looking for a full, i.e., per-turbed, (time-independent) contact transformationξh = Tfull(ξEOB) = T0(ξEOB) + µ1Tµ1(ξEOB) thattransforms Hh

full(ξh) into HEOBfull (ξEOB), i.e., such that

Hhfull[Tfull(ξEOB)] = HEOB

full (ξEOB) . (5.11)

Rewriting the full transformation Tfull as the composi-tion T ′ T0 of the known unperturbed (tidal-free) con-tact transformation ξ0h = T0(ξEOB) mapping Hh

0 (ξ0h) intoHEOB

0 (ξEOB) with an unknown near-identity additionaltransformation, ξh = T ′(ξ0h) = ξ0h + µ1Gµ1(ξ

0h), ξ0h

[where f, g denotes a Poisson bracket and whereµ1Gµ1(ξ

0h) is the first-order generating function associ-

ated with the canonical transformation T ′], and expand-ing all functions in Eq. (5.11) into unperturbed plus tidalcontributions (Hh = Hh

0 + δHh, T = (1 + δT ′) T0,HEOB = HEOB

0 + δHEOB), leads to the condition

[δHh(ξ0h) + δG(ξ0h), Hh(ξ0h)

]ξ0

h=T0(ξEOB)

= δHEOB(ξEOB) , (5.12)

where δG(ξ0h) = µ1Gµ1(ξ0h).

In general, δG(ξ0h) is part of the unknown functionsthat must be looked for when writing the condition inEq. (5.12). However, another simplifying fact occurs inthe case where one focusses on circular motions: Thesupplementary term δG,Hh happens to vanish. In-deed, δG(ξ0h) is a scalar function and the Poisson bracketδG,Hh is equal to the time derivative of δG(ξ0h) alongthe Hh-dynamical flow, which clearly vanishes along cir-cular motions. This allows one to conclude that, alongcircular motions, we have the simple condition

[δHh(ξ0h)

]circξ0

h=T0(ξEOB)

=[δHEOB(ξEOB)

]circ, (5.13)

where the left-hand side is, in principle, fully known.

C. Link between the circular tidal action and thetidal contribution to the EOB A potential

Let us now evaluate the right-hand side of Eq. (5.13).When restricting the definition of Eqs. (5.4), (5.5) ofthe EOB Hamiltonian to circular motions, the terms(pEOB

r )2/B and Q(rEOB, pEOB) disappear (because one

works with a gauge-reduced Q which vanishes withpEOB

r ). As a consequence, HcircEOB(rEOB, J) only de-

pends on the A potential. The difference, δHcircEOB ≡

HcircEOB[rEOB, J, Afull] − Hcirc

EOB[rEOB, J, A0], can then besimply computed by varying A (Afull = A0 + δA) withinHcirc

EOB[A]. To write explicitly the result of this varia-tion, it is convenient to work with dimensionless vari-ables. We can replace the two phase-space variablesrEOB, pEOB

ϕ ≡ J that enter HcircEOB by their dimension-

less counterparts

u ≡ GM

c2 rEOB≡ G(m1 +m2)

c2 rEOB, (5.14)

and

j ≡ c J

GM µ≡ c J

Gm1m2. (5.15)

In terms of these variables, the explicit expression of[HEOB

full

]circreads

[HEOB

full (u, j)]circ

= M c2√

1 + 2 ν(−1 +

√A(u)(1 + j2 u2)

). (5.16)

Varying A(u) in Eq. (5.16) then yields the following ex-plicit expression for the right-hand side of Eq. (5.13):

19

[δHEOB(u, j)

]circ=

1

2M ν c2

√√√√ 1 + j2 u2

A(u)[1 + 2 ν

(− 1 +

√A(u)(1 + j2 u2)

)] δA(u) . (5.17)

In addition, one must take into account the constraintcoming from the reduction to circular motions, namely,from pEOB

r = −∂HEOB/∂ rEOB, the fact that ∂u[A(u)(1+j2 u2)] = 0, i.e. the fact that j2 is the following functionof u (using a prime to denote the u-derivative):

j2 = j2circ(u) ≡ −A′(u)

(u2A(u))′. (5.18)

Note that this relation depends on the value of the radialpotential A(u). If one is considering the full, tidally-perturbed circular motions one must use Afull(u) =A0 + δA in Eq. (5.18). On the other hand, as we arenow interested in considering the (first-order) tidal per-turbations δHh and δHEOB, and their link in Eq. (5.13),we can evaluate δHEOB

circ with sufficient accuracy by re-

placing in the coefficient of δA(u), on the right-hand sideof Eq. (5.17), A(u) and j2 by their unperturbed, tidal-free expressions A0(u) and j2A0

(u) (obtained by replacingA → A0 on the right-hand side of Eq. (5.18). [This re-mark applies to several other results below; notably Eqs.(5.20) and (5.22)].

Combining our results of Eqs. (5.9), (5.13) and (5.17),we finally get a very simple link between the tidal vari-ation of the harmonic-coordinate Lagrangian δL(yh, vh)and the corresponding tidal variation δA(u) of the EOBA potential, namely,

δA(u) = − 2

M ν c2

√F (u) [δL(yh, vh)]

circrh=T0(u) , (5.19)

where

F (u) ≡[

A(u)

1 + j2 u2

(1 + 2 ν

(−1 +

√A(u)(1 + j2 u2)

))]circ

A=A0

. (5.20)

Here, the superscript “circ” means that j2 must be re-placed by j2circ(u), Eq. (5.18). (Note that the replacementA → A0 indicated as a subscript must be done both inthe explicit occurrence of A in Eq. (5.20) and in the def-inition in Eq. (5.18) of j2circ(u)). Finally, if we introducethe short-hand notation

A(u) ≡ A(u) +1

2uA′(u) , (5.21)

F (u), Eq. (5.20), can be rewritten in the explicit form

F (u) = A(u)

1 + 2 ν

−1 +

A(u)√A(u)

, (5.22)

which is valid along circular orbits, and applies for anyrelevant (exact or approximate) value of the A potential.On the other hand, as we computed δL only to the 2PNfractional accuracy, it is sufficient to use a value of F (u)which is also only fractionally 2PN-accurate. One mightthink a priori that this would mean using for A(u) inEq. (5.22) the tidal-free approximation A0(u) truncatedat the 2PN order, namely A2PN

0 (u) = 1 − 2 u + 2 ν u3.However, the contribution 2 ν u3 = 2 ν(GM/(c2 rEOB))3

is O(1/c6) compared to one, which is the leading-ordervalue of F (u), which starts as F (u) = 1 + O(u) = 1 +

O(1/c2). The same consideration applies to A(u). [Thesituation would have been different if F (u) had been,say, ∝ A′(u).] This means that, at the 2PN fractionalaccuracy, we can use the value of F (u) obtained fromthe leading-order, “Schwarzschild-like” value of A0(u),

namely A1PN0 (u) = 1−2 u. The corresponding A function

is then: A1PN0 (u) = 1− 3 u, so that

F 2PN(u) = (1 − 3 u)

[1 + 2 ν

(− 1 +

1− 2 u√1− 3 u

)].

(5.23)Consistently with the fractional 2PN accuracy, and re-membering, that u = O(1/c2), we could as well usethe 2PN-accurate series expansion of Eq. (5.23), sayF 2PN(u) = 1 + f1(ν)u + f2(ν)u

2 + O(u3). However,it is better to retain the information contained in Eq.(5.23) that, in the test-mass limit ν → 0 (where A0(u) →1 − 2 u), the exact value of F (u) becomes 1 − 3 u (seelater).

There remains only one missing piece of informationto be able to use our result in Eq. (5.19) for computingthe various tidal contributions to A(u). We need to workout the explicit form of the unperturbed transformationT0 between rEOB and rh.

A first method for getting the transformation T0 (at

20

2PN) is to compose the transformation ξ0h → ξADM (ob-tained at 2PN in Ref. [46], and at 3PN in Ref. [50]) withthe transformation ξADM → ξEOB (obtained at 2PN inRef. [16], and at 3PN in Ref. [18]). For our presentpurpose, it is enough to restrict these transformations tothe circular case, i.e. to transformations rh → rADM andrADM → rEOB.

The transformation h → ADM starts at 2PN,i.e., yh

A = xADMA + c−4 Y 2PN

A (xADM,pADM), withY 2PN

A (xADM,pADM) given, e.g., in Eq. (4.5) of Ref. [50].Its circular, and center-of-mass, reduction (with n12 ·pA = 0, p1 = −p2 ≡ p, and (p/µ)2 = GM/r12+O(1/c2))yields at 2PN

rh12 = rADM

12

[1 +

(1

4+

29

8ν

)(GM

c2 r12

)2]. (5.24)

On the other hand the transformation ADM → EOBstarts at 1PN. To determine the corresponding radialtransformation rADM

12 → rEOB, one could think of us-ing Eq. (6.22) of Ref. [16]. However, this equationneeds to be completed by the knowledge of the circularitycondition relating (pADM/µ)2 to GM/rADM

12 at the 1PNlevel included. This 1PN-accurate circularity conditioncan, e.g., be obtained from combining the 1PN-accuraterADM = rADM(j) relation given in Ref. [51] (see be-low), with the fact that (setting uADM ≡ GM/(c2 rADM))(pADM/(µc))

2 = j2 u2ADM. This yields (pADM/(µc))

2 =uADM + 4 u2

ADM, and therefrom the relation betweenrADM and rEOB.

Another method (which we have checked to give thesame result) for determining the rADM

12 → rEOB transfor-mation does not need to use Eq. (6.22) of Ref. [16]. Itconsists of directly eliminating the dimensionless angularmomentum j between the two relations rADM = rADM(j)and rEOB = rEOB(j). The former relation was derivedat 3PN in Ref. [51] and reads, at 2PN,

rADM12 =

GM

c2j2[1− 4

j2− 1

8(74− 43 ν)

1

j4

], (5.25)

while the latter one is obtained by inverting the 2PN-accurate version of Eq. (5.18), namely, using A2PN(u) =1− 2 u+ 2 ν u3:

1

j2=u(1− 3 u+ 5 ν u3)

1− 3 ν u2. (5.26)

Inserting Eq. (5.26) into Eq. (5.25) yields (at 2PN)

GM

c2 rADM12

= u

[1 + u+

(5

4− 19

8ν

)u2

]. (5.27)

Then, combining Eq. (5.27) and Eq. (5.24) yields thelooked for transformation rEOB → rh

12, at 2PN accuracy,

rh12 +

GM

c2= rEOB

(1 + 6 ν

(GM

c2 rEOB

)2), (5.28)

or, setting uh ≡ GM/(c2 rh12) by analogy with u ≡

GM/(c2 rEOB),

uh =u

1− u(1 − 6 ν u2) . (5.29)

We have written the transformation of Eqs. (5.28), (5.29)so as to exhibit the exact form of the transformationrh → rEOB in the extreme mass ratio limit ν → 0, namelyrh = rEOB −GM/c2 +O(ν).

Summarizing: The (first-order) tidal contributionδA(u) = µ1Aµ1(u) to the main EOB radial po-tential, associated with any tidal parameter µ1 (=

µ(2)1 , µ

(2)2 , µ

(3)1 , . . .), is given in terms of the correspond-

ing harmonic-coordinate tidal contribution to the ac-tion δL(yh, vh) = µ1Lµ1(yh, vh), for circular motion, byEq. (5.19), where F (u) is given (at 2PN) by Eq. (5.23),and where the transformation between the harmonicradial separation rh

12 and the EOB radial coordinaterEOB ≡ GM/(c2 u) is given by Eqs. (5.28) or (5.29).

VI. EOB DESCRIPTION OF TIDAL ACTIONS

A. Tidal actions for comparable-mass systems

We have explained in the previous section how to con-vert each contribution ∼ µ1 Lµ1(yh, vh) to the (reduced)tidal action into a corresponding additional contributionµ1Aµ1(u) to the main EOB radial potential A(u). Forinstance, if we consider the dominant tidal parameter,

i.e. the electric quadrupolar one, µ(ℓ=2)1 (or µ

(ℓ=2)2 , af-

ter exchanging 1 ↔ 2), the combination of the result ofEq. (4.2) for the associated Lagrangian, with Eq. (5.19)yields

µ(2)1 A

µ(2)1

(u) = − 1

2 c2µ

(2)1

Mν

√F (u)

dτ1dt

[Gαβ Gαβ ]1 .

(6.1)In other words, apart from a (negative) numerical coeffi-

cient, and the rescaled tidal parameter µ(2)1 /(Mν) (where

Mν = µ = m1m2/(m1 +m2) is the reduced mass of thesystem), the corresponding tidal contribution to A(u) is

the product of three factors:√F (u), dτ1/dt and the ge-

ometrical invariant associated with the considered tidalparameter, e.g., [Gαβ G

αβ ]1 for the electric quadrupolealong the first worldline. In addition, two of these fac-tors, dτ1/dt and the geometrical invariant, must be re-expressed as functions of the EOB coordinates by usingEq. (5.28).

Let us start by applying this procedure to the dom-inant tidal action term: the electric-quadrupole one inEq. (6.1). We have given above, in Eq. (4.14), the 2PN-accurate value of J2e ≡ [Gαβ G

αβ ]1 in harmonic coordi-nates. Using the transformation of Eq. (5.29) to replace1/rh

12 in terms of 1/rEOB leads to

J(circ)2e =

6M2X22

r6EOB

[1 + ǫ2

(X1 + 3)M

rEOB

21

+ǫ4M2

28r2EOB

(295X21 − 7X1 + 336)

]. (6.2)

In addition the reexpression of the time-dilation factordτ1/dt, Eq. (4.19), in terms of 1/rEOB yields

dτ1dt

=1

Γ1= 1− 1

2(X1 − 1)(X1 − 3)uǫ2

+3

8u2(X1 − 1)(X3

1 − 3X21 + 3X1 + 3)ǫ4 . (6.3)

Their product yields the electric-quadrupole tidal La-

grangian (stripped of its prefactor 14 µ

(2)1 ) in EOB co-

ordinates, at the 2PN accuracy, namely

G2ab

dτ1dt

=J

(circ)2e

Γ1=

6(X1 − 1)2u6

M4L2e , (6.4)

where

L2e = 1− 1

2u(X2

1 − 6X1 − 3)ǫ2 (6.5)

+u2

56(21X4

1 − 112X31 + 744X2

1 + 238X1 + 357)ǫ4 .

Adding the further factor√F (u), as well as the prefac-

tor, leads to the corresponding contribution to the EOBA potential, namely

µ(2)1 A

µ(2)1

(u) = A(2)LO1 electric(rEOB) A

(2)1 electric(u) , (6.6)

where

A(2)LO1 electric(rEOB) = − 3G2

c2µ

(2)1 M

ν

X22

r6EOB

, (6.7)

and

A(2)1 electric(u) =

√F (u) L2e = 1 + α2e

1 u+ α2e2 u2 +O(u3) ,

(6.8)with

α2e1 =

5

2X1 , (6.9)

α2e2 =

337

28X2

1 +1

8X1 + 3 . (6.10)

The leading-order (i.e., Newtonian-level) A potential ofEq. (6.7) is equivalent to Eqs. (1.6) and (1.7) above (i.e.,Eqs. (23), (25) of Ref. [5]), using the link

Gµ(ℓ)A =

1

(2ℓ− 1)!!2 k

(ℓ)A R2ℓ+1

A . (6.11)

The term of order u (i.e., 1PN) in the relativistic ampli-

fication factor A(2)1 electric(u), Eq. (6.8), coincides with the

result computed some time ago (see Eq. (38) in Ref. [5]).

By contrast, the (2PN) term of order u2 in A(2)1 electric(u) is

the main new result of our present work. Let us discussits properties.

Similarly to the 1PN coefficient α2e1 = 5

2 X1, which waspositive, and monotonically increasing (from 0 to 5/2) asX1 ≡ m1/M varies between 0 and 1, the 2PN coefficientα2e

2 is also positive, and increases as X1 varies between0 and 1. When X1 = 0 (i.e. in the limit m1 ≪ m2), α

2e2

takes the value + 3, while when X1 = 1 (i.e., in the limitm1 ≫ m2), it takes the value

α2e2 (X1 = 1) =

849

56= 15.16071429 . (6.12)

Note that this is about 5 times larger than its value whenX1 = 0. Of most interest (as neutron stars are expectedto have rather similar masses∼ 1.4M⊙) is the equal-massvalue of α2e

2 , which is

α2e2

(X1 =

1

2

)=

85

14= 6.071428571 . (6.13)

In other words, the distance-dependent amplificationfactor of the electric quadrupole reads, in the equal-masscase

[A

(2)1 electric(u)

]equal-mass

= 1 +5

4u+

85

14u2 +O(u3)

= 1 + 1.25 u+ 6.071429 u2

+O(u3) . (6.14)

We will comment further on these results for A(2)1 electric(u)

and on the recent comparisons between numerical simu-lations and the EOB description of tidal interactions be-low. For the time being, let us give the correspondingresults of our analysis for some of the sub-leading tidalinteractions.

The EOB-coordinate value of the electric octupole in-

variant, J(circ)3e , Eq. (4.17), reads

J(circ)3e =

90X22M

2

r8EOB

[1 + ǫ2(6X1 + 1)

M

rEOB

+ǫ4M2

3r2EOB

(83X21 + 14X1 + 17)

]. (6.15)

Its corresponding action (stripped of its prefactor) is

G2abc

dτ1dt

=J

(circ)3e

Γ1=

90X22u

8

M6L3e (6.16)

with

L3e = 1− 1

2(X2

1 − 16X1 + 1)uǫ2 (6.17)

+1

24(9X4

1 − 108X31 + 994X2

1 − 56X1 + 73)u2ǫ4

while the corresponding contribution to the EOB A po-tential reads

µ(3)1 A

µ(3)1

(u) = A(3)LO1 electric(rEOB) A

(3)1 electric(u) , (6.18)

22

where

A(3)LO1 electric(rEOB) = − 15G2

c2µ

(3)1 M

ν

X22

r8EOB

, (6.19)

and

A(3)1 electric(u) =

√F (u) L3e = 1 + α3e

1 u+ α3e2 u2 +O(u3) ,

(6.20)with

α3e1 =

15

2X1 − 2 , (6.21)

α3e2 =

110

3X2

1 −311

24X1 +

8

3. (6.22)

Here, both results in Eqs. (6.21) and (6.22) are new.Note that, contrary to the quadrupolar case where α1 and

α2 were always both positive (so that A(2)1 electric(u) was al-

ways an amplification factor) the electric-octupole factor

A(3)1 electric(u) is smaller than 1 (for large separations) when

X1 <415 ≃ 0.2667. Moreover, while the X1-variation of

α3e1 is monotonic (going from − 2 to 11

2 as X1 increases

from 0 to 1), α3e2 (X1) first decreases from α3e

2 (0) = 83 =

2.666667 to α3e2 (Xmin

1 ) = 42853/28160 = 1.521768 as X1

increases from 0 toXmin1 = 311/1760 = 0.1767046, before

increasing as X1 goes from Xmin1 to 1, to reach the final

value α3e2 (1) = 211/8 = 26.375 for X1 = 1. Note, how-

ever, that when (as expected) the two masses are nearly

equal the factor A(3)1 electric(u) is an amplification factor.

In particular, its equal-mass value is

[A

(3)1 electric(u)

]equal-mass

= 1 +7

4u+

257

48u2 +O(u3)

= 1 + 1.75 u+ 5.354167 u2

+O(u3) (6.23)

which is similar to its corresponding quadrupolar coun-terpart, Eq. (6.14).

Let us finally give the corresponding results for themagnetic quadrupole and time-differentiated electricquadrupole. For the magnetic quadrupole (at the 1PNfractional accuracy), we found

1

4H2

ab ≡ J(circ)2m (6.24)

=18X2

2M3

r7EOB

[1 + ǫ2

M

3rEOB(3X2

1 +X1 + 12)

],

1

4H2

ab

dτ1dt

≡ 18X22

M4u7L2m , (6.25)

L2m = 1 +1

6(X1 + 3)(3X1 + 5)uǫ2 , (6.26)

A(2)1 magnetic(u) =

√F (u) L2m = 1 + α2m

1 u+O(u2) ,

(6.27)

with

α2m1 = X2

1 +11

6X1 + 1 . (6.28)

Here α2m1 (X1) is always positive, and monotonically in-

creases from α2m1 (0) = 1 to α2m

1 (1) = 236 = 3.833333, its

equal-mass value being α2m1

(12

)= 13

6 = 2.166667.

Finally, for the time-differentiated electric quadrupole,we got

G2ab = J

(circ)

2e=

18X22M

3

r9EOB

[1 + ǫ2(X2

1 + 2)M

rEOB

],

(6.29)

G2ab

dτ1dt

=18X2

2

M6u9 L2e , (6.30)

L2e = 1 +1

2uǫ2(X2

1 + 4X1 + 1) , (6.31)

A(2)

1G(u) =

√F (u) L2e = 1 + α2e

1 u+O(u2) , (6.32)

with

α2e1 =

1

2(X1 + 2)(2X1 − 1) . (6.33)

B. Tidal actions of a tidally-deformable test mass

One of the characteristic features of the EOB formal-ism for point-mass systems is the natural incorporationof the exact test-mass limit ν → 0. Indeed, in thislimit the effective metric in Eq. (5.2) describing therelative dynamics reduces to the Schwarzschild metric:

limν→0 A(u) = 1− 2 u =(limν→0 B(u)

)−1. Let us study

the test-mass limit of tidal effects, with the aim of incor-porating it similarly in their EOB description. Whenconsidering the nonminimal worldline action of parti-cle 1, the simplest test-mass limit to study is the limitm1/m2 → 0. [When considering tidal effects within body2, the permutation 1 ↔ 2 of our results below allow themto describe the limit m2/m1 → 0. We leave to futurework a study of the limit m2/m1 → 0, when consid-ering tidal effects taking place within body 1.] In thelimit investigated here, one is considering a tidally de-

formable test-mass (m1, µ(ℓ)1 , . . .) moving around a large

mass m2 ≫ m1. The effective action of body 1 is thenexactly obtained by evaluating the A = 1 contributionof the general (two-body) effective action of Eq. (2.12)within the background metric generated by the (non-tidally deformable) large mass m2, at rest, i.e. withina Schwarzschild metric of mass m2. The latter reads

ds2(m2) = −(

1− 2Gm2

c2 rs

)c2 dt2 +

dr2s1− 2 G m2

c2 rs

+r2s(dθ2 + sin2 θ dϕ2) (6.34)

23

in “Schwarzschild”, or areal, coordinates, and

ds2(m2) = −1− G m2

c2 rh

1 + G m2

c2rh

c2 dt2 +1 + G m2

c2 rh

1− G m2

c2rh

dr2h

+

(rh +

Gm2

c2

)2

(dθ2 + sin2 θ dϕ2) (6.35)

in harmonic coordinates: rh = rs −Gm2/c2. As a check

on the results below (and on our codes), we have com-puted them both in Schwarzschild coordinates and in har-monic ones.

The geometrical invariants J2e = G2ab, etc., take the

following values in this Schwarzschild limit, and whenconsidering as above circular motions (we again set Gand c to one for simplicity):

G(S)ab

2 = J(S)2e =

6m22(m

22 + r2h −m2rh)

(rh − 2m2)2(rh +m2)6

∼ 6m22

r6h

[1− 3m2

rh+

12m22

r2h+ . . .

]

=6u6

S

m42

1

(1− 3uS)

[1 +

3u2S

(1− 3uS)

]

=6u6

S

m42

[1 + 3uS

(1− 2uS)

(1 − 3uS)2

], (6.36)

1

4H

(S)ab

2 = J(S)2m =

18m32(rh −m2)

(rh − 2m2)2(rh +m2)6

∼ 18m32

r7h

[1− 3m2

rh+

11m22

r2h+ . . .

]

=18u7

S

m42

[1 +

uS(4 − 9uS)

(1− 3uS)2

], (6.37)

G(S)abc

2 = J(S)3e =

30m22(rh −m2)(2m

22 + 3r2h − 3m2rh)

(rh − 2m2)2(rh +m2)9

∼ 90m22

r8h

[1− 7

m2

rh+

98

3

m22

r2h+ . . .

]

=90u8

S

m62

(1− 2uS)

(1− 3uS)

[1 +

8u2S

3(1− 3uS)

],

(6.38)

G(S)ab

2 = J(S)

2e=

18m32(rh −m2)

2

(rh − 2m2)2(rh +m2)9

=18m3

2

r9h

[1− 7

m2

rh+ 32

m22

r2h+ . . .

]

=18u9

S

m62

(1− 2uS)2

(1− 3uS)2, (6.39)

where uS ≡ Gm2/(c2 rs). We have indicated above the

expansions in powers of the inverse harmonic radius rh aschecks of our 2PN-accurate results, written in harmoniccoordinates; see Eqs. (4.14), (4.16)–(4.18).

In the following, we shall focus on the transformationof the exact test-mass geometrical invariants above into

corresponding contributions to the EOB A potential. Asexplained previously, Eqs. (5.19), (6.1), apart from theuniversal prefactor − 2/(M ν c2) and the specific originaltidal coefficient multiplying the considered geometrical

invariant (such as 14 µ

(2)1 for the electric quadrupole), the

contribution toA(u) associated with some given invariantis obtained by multiplying it by two extra factors: (i)the time-dilation factor dτ1/dt and (ii) the EOB-rooted

factor√F (u). Let us discuss their values in the test-mass

limit m1 ≪ m2 that we are now considering.

The first factor is the square-root of

(dτ1dt

)2

= 1− 2Gm2

c2 rs− 1

c2r2s

(dϕ

dt

)2

. (6.40)

Denoting, as above, uS ≡ Gm2/(c2 rs), and us-

ing the well-known Kepler law for circular orbits inSchwarzschild coordinates, Ω2 = Gm2/r

3s , simply yields

(dτ1dt

)test-mass

circ

=√

1− 3 uS . (6.41)

The exact test-mass limit of the second factor is ob-tained by taking the limit ν → 0 in the exact expressionof Eq. (5.22). In this limit, A(u) → 1 − 2 u, so that

A(u) → 1− 3 u, and

(√F (u)

)test-mass

circ=√

1− 3 u . (6.42)

In addition, as the EOB coordinates reduce toSchwarzschild coordinates in the test-mass limit ν → 0,and M = m1 +m2 → m2, we have simply

uS ≡Gm2

c2 rs→ u ≡ GM

c2 rEOB. (6.43)

In other words, the two extra factors in Eqs. (6.41),(6.42) become both equal to

√1− 3 u. As a consequence

the A contribution corresponding to the various geomet-rical invariants of Eqs. (6.36)–(6.39) is obtained (apartfrom a constant prefactor) by multiplying these invari-

ants by(√

1− 3 u)2

= 1 − 3 u = 1 − 3 uS. Including the

universal factor −2/(M ν c2) and the various tidal coef-

ficients 12

1ℓ! µ

(ℓ)1 , 1

2ℓ

ℓ+11ℓ!

σ(ℓ)1

c2 , . . . (as well as the factor

4 in H2ab = 4J2m) yields the following exact, test-mass

contributions

µ(2)1 Atest-mass

µ(2)1

(u) = − 3G2

c2µ

(2)1

m1

(m2)2

r6EOB

(1 +

3 u2

1− 3 u

),

(6.44)

µ(3)1 Atest-mass

µ(3)1

(u) = − 15G2

c2µ

(3)1

m1

(m2)2

r8EOB

(1− 2 u)×

×(

1 +8

3

u2

1− 3 u

), (6.45)

24

σ(2)1 Atest-mass

σ(2)1

(u) = − 24G3

c4σ

(2)1

m1

(m2)3

r7EOB

1− 2 u

1− 3 u,

(6.46)

µ′(2)1 Atest-mass

µ′(2)1

(u) = − 9G3

c4µ′(2)1

m1

(m2)3

r9EOB

(1 − 2 u)2

1− 3 u.

(6.47)One easily sees that the various exact, test-mass am-

plification factors A(u) exhibited here are compatiblewith the X1 → 0 limit of the 2PN-expanded ones ∼1 + α1u+ α2 u

2 +O(u3) derived above.

C. Light-ring behavior of test-mass tidal actions

A striking feature of all the amplification factorspresent in Eqs. (6.44)–(6.47), such as

A(2) test-mass1 electric (u) = 1 + 3

u2

1− 3 u, (6.48)

is that they all formally exhibit a pole ∝ 1/(1 − 3 u)mathematically located at 3 u = 1, i.e. correspondingto formally letting particle 1 tend to the last unstablecircular orbit, located at 3Gm2/c

2 (“light-ring” orbit).This behavior has a simple origin.

The invariant that is simplest to consider in order tosee this is J2e = G2

ab. From Eq. (4.3) its covariant ex-pression reads

G2ab = Rαµβν R

α β•κ • λ u

µ uν uκ uλ . (6.49)

Let us study its mathematical behavior in the formallimit where particle 1 tends to the light-ring orbit.Using the language of Special Relativity, we considerthe Schwarzschild coordinates as defining a “lab-frame.”With respect to this lab-frame, particle 1 becomes ultra-relativistic as it approaches the light ring. More pre-cisely, near the light ring the lab-frame components ofthe 4-velocity uµ = (dt/dτ1)(c, v

i) tend towards infin-ity proportionally to dt/dτ1 = Γ1 = 1/

√1− 3 u, while

the lab-frame components of Rαµβν (and of the met-ric) stay finite. As G2

ab is quartic in the lab-frame com-ponents of uµ, it will tend towards infinity like Γ4

1 =(dt/dτ1)

4 = (1 − 3 u)−2. The corresponding contribu-tion to A(u) is obtained by multiplying G2

ab by the factor

(dτ1/dt)2 = Γ−2

1 = (1−3 u)+1, which reduces the blow-upof G2

ab to the milder (1−3 u)−2+1 = (1−3 u)−1 behaviorthat is apparent in Eqs. (6.44) or (6.48).

A different way of phrasing this result uses the lawof transformation of the electric and magnetic compo-nents of the Weyl tensor, Gab and Hab, under a boost.Using, for instance, the fact that, under a boost with ve-locity β = tanhϕ in the x direction, the complex ten-sor Fab = Gab + iHab undergoes a complex rotationof angle ψ = i ϕ in the yz plane [52], one easily findsthat the transverse traceless components of Fab (in theyz plane) acquire, under such a boost, a factor of order

cos2 ψ = cosh2 φ = (1− β2)−1 ≡ Γ21. Because of the spe-

cial structure of the tensor Fab ∝ diag (−1,−1, 2), withthe third axis z labelling the radial direction, this rea-soning shows that boosts in the radial (z) direction leaveFab invariant. However, we are mainly interested here inboosts in a “tangential” direction, say x, associated withthe fast motion of a circular orbit, and therefore orthogo-nal to the radial direction, which do introduce a factor Γ2

1

in some of the boosted components of Fab. For complete-ness, let us indicate that because of this special structureof Fab, the invariant J2e = G2

ab for general, non-circular

orbits is equal to

J2e = G2ab =

6m22

r6s

(1 + 3u2

tg + 3u4tg

), (6.50)

where u2tg ≡ r2s((uθ)2 + sin2 θ(uφ)2) is the square of the

part of the 4-velocity uµ that is tangent to the sphere.[The radial component of the 4-velocity brings no contri-bution to J2e.]

The behavior near the light ring of the magnetic-quadrupole invariant J2m = 1

4H2ab is understood in the

same way as that of J2e = G2ab. Concerning the other

invariants, one can note that J3e = G2abc can be written

as the sum

J3e = G2abc = Cαβγ C

αβγ +1

3 c2J2e (6.51)

where

Cαβγ = Symαβγ ∇αRβµγν uµ uν , (6.52)

and

J2e = G2ab = Gαβ G

αβ (6.53)

with

Gαβ = uλ∇λ Rαµβν uµ uν . (6.54)

Similarly to G2ab, Eq. (6.49), the term C2

αβγ in Eq. (6.51)is quartic in uµ and is therefore expected to blow up likeΓ4

1. On the other hand, though Gαβ , Eq. (6.54), is cubicin uµ, it only blows up like Γ2

1 (so that J2e ∼ Γ41 and J3e ∼

C2 +J2e ∼ Γ41) because of the special geodetic-precession

properties of the proper-time derivative operator∇/dτ =uλ∇λ (see, e.g., Sec. 3.6 of Ref. [53]).

D. A suggested “resummed” version ofcomparable-mass tidal actions

Having understood that the formal pole-like behav-ior, ∼ (1 − 3 u)−1, in the test-mass limit of the electric-quadrupole A potential is linked to simple boost proper-ties of Gab near the light-ring orbit, and knowing that theEOB formalism predicts the existence of a formal analogof the usual Schwarzschild light ring at the EOB dimen-sionless radius rLR ≡ 1/uLR, defined as the solution of

A(uLR) = 0 , (6.55)

25

with A(u) defined in Eq. (5.21), it is natural to expectthe (unknown) exact two-body version of the electric-quadrupole A potential to mathematically exhibit ananalogous pole-like behavior of the form ∼ (1− rLR u)

−1.As we shall discuss elsewhere, such a mathematical be-havior, linked to considering (within the EOB-simplifyingapproach advocated in Ref. [18]) what would happen ifone formally considered (unstable) circular orbits withu → uLR, does not mean that there is a real physicalsingularity in the EOB dynamics near u = uLR, butit indicates that higher-than-2PN contributions to the

electric-quadrupole amplification factor A(2)1 electric(u) =

1 + α2e1 u+ α2e

2 u2 + α2e3 u3 + · · · will probably be slowly

convergent, and will tend to amplify further the corre-sponding tidal interaction. Such an extra amplificationmight, for instance, be physically important in the lastorbits of comparable-mass neutron-star binaries (whichwill reach contact for values of u smaller than uLR).

This leads us to suggest that a more accurate value (foru < uLR) of the electric-quadrupole amplification factoris the following “resummed” version of Eq. (6.8):

A(2)1 electric(u) = 1 + α2e

1 u+ α2e2

u2

1− rLR u, (6.56)

where α2e1 and α2e

2 are given by Eqs. (6.9) and (6.10), andwhere rLR ≡ 1/uLR is the solution of Eq. (6.55). Simi-lar resummed versions of the other amplification factorscan be defined by incorporating in their PN-expandedversions the formal light-ring behaviors exhibited by theexact test-mass results of Eqs. (6.44)–(6.47).

Let us finally discuss several possible approximate val-ues for rLR in the proposed Eq. (6.56). The simplest ap-proximation consists of using the “Schwarzschild” valuerSLR = 3. However, a better value might be obtained by

taking a solution of Eq. (6.55) that incorporates morephysical effects. This might require solving Eq. (6.55)numerically, with A(u) being the full A potential (con-taining both Pade-resummed two-point-mass effects andthe various tidal contributions). In order to have a feel-ing for the modification of rLR brought by incorporatingthese changes, let us consider solving Eq. (6.55) when us-ing the following approximation to the full A potential:

Aapprox(u) = 1− 2 u+ 2 ν u3 − κu6 (6.57)

where

κ = κ(2)1 + κ

(2)2 = 2 k

(2)1

m2

m1

(R1 c

2

G(m1 +m2)

)5

+2 k(2)2

m1

m2

(R2 c

2

G(m1 +m2)

)5

. (6.58)

Here, the term + 2 ν u3 is the 2PN-accurate point-massmodification of A(u), while the term −κu6 is the leading-order tidal modification. Note that they have oppositesigns. The corresponding expression of A(u) reads

Aapprox(u) = 1− 3 u+ 5 ν u3 − 4 κu6 . (6.59)

The corresponding value of uLR ≡ 1/rLR is the solutionclose to 1/3 of the equation

uLR =1

3

[1 + 5 ν u3

LR − 4 κu6LR

]. (6.60)

If we could treat both ν and κ as small deformationparameters, this would imply that, to first order inthese two deformation parameters, the value of uLR(ν, κ)would be obtained by inserting the leading-order valueuLR ≃ 1/3 in the right-hand side of Eq. (6.60). Thiswould yield

uLR(ν, κ) =1

3

[1 +

5

33ν − 4

36κ+O(ν2, νκ, κ2)

],

(6.61)and

rLR(ν, κ) = 3

[1− 5

33ν +

4

36κ+O(ν2, νκ, κ2)

].

(6.62)Note that while comparable-mass corrections (∝ ν) havethe effect of decreasing rLR, tidal ones (∝ κ) have theopposite effect of increasing rLR. Let us focus on the tidaleffects, and consider the equal-mass case with R1 = R2

and k(2)1 = k

(2)2 . One has a first order increase of rLR

equal to

δtidal rLR ≃ 16 k(2)1

(R1 c

2

6Gm1

)5

= 16 k(2)1

1

(6 C1)5, (6.63)

where C1 ≡ Gm1/(c2R1) denotes the common compact-

ness of the two neutron stars. This simple approximateanalytical formula shows that δtidal rLR is very sensitiveto the value of the compactness of the neutron star. IfC1 = 1/6 = 0.166667, i.e., R1 = 6Gm1/c

2 (roughly cor-responding to a radius of 12 km for a 1.4M⊙ neutron

star), then δtidal rLR = 1.44 (k(2)1 /0.09) will be of order

1 [the value k(2)1 = 0.09 being typical for C1 = 1/6;

see, e.g., Table II in Ref. [5]]. On the other hand, if

R1 ≡ R1 c2/(Gm1) is slightly smaller than 6, δtidal rLR

will quickly become much smaller than 1, while if R1

is slightly larger than 6, δtidal rLR will quickly becomeformally large (thereby invalidating the first-order ana-lytical estimate of Eq. (6.63), which assumed δ rLR ≪ 3).These rough estimates indicate that in many cases, tidaleffects on rLR will be quite important and will signifi-cantly increase the numerical value of rLR. Note thatan increased value of rLR will, in turn, increase theeffect of the conjectured resummed 2PN contribution

α2e2 u2/(1− rLR u) to A

(2)1 electric(u).

VII. SUMMARY AND CONCLUSIONS

Using an effective action technique, we have shownhow to compute the additional terms in the reduced

(Fokker) two-body Lagrangian L(y1,y2, y1, y2) that are

26

linked to tidal interactions. Thanks to a general prop-erty of perturbed Fokker actions [explained at the endof Sec. II, see Eq. (2.20)], the additional tidal termsare correctly obtained (to first order in the tidal per-turbations) by replacing in the complete, unreduced ac-tion S[gµν ; y1, y2] the gravitational field gµν by the so-lution of Einstein’s equations generated by two struc-tureless point masses m1,y1;m2,y2. This allowed usto compute in a rather straightforward manner the re-duced tidal action at the 2PN fractional accuracy by us-ing the known, explicit form of the 2PN-accurate two-point-mass metric [36–39]. The main technical subtletyin this calculation is the regularization of the self-field ef-fects associated with the computation of the various non-minimal tidal-action terms ∼

∫dτ(Rαµβν u

µ uν)2 + . . .,where, e.g., Rαµβν(x; y1, y2) is to be evaluated on oneof the worldlines that generate the metric gµν (so thatRαµβν(y1; y1, y2) is formally infinite). We explainedin detail (in Sec. III) one (algorithmic) way to per-form this regularization, using Hadamard regularization(which is equivalent to dimensional regularization at the2PN level). We then computed the regular parts ofthe brick potentials that parametrize the 2PN metric,from which we derived the regularized values of sev-eral of the geometrical invariants entering the nonmin-imal worldline tidal action terms. [See Eqs. (4.4)–(4.10)for the 2PN-accurate Lagrangians (for general orbits) ofthe three leading tidal terms (electric quadrupole, elec-tric octupole and magnetic quadrupole)]. We then fo-cussed on the most physically useful information con-tained in these actions, namely the corresponding con-tributions to the EOB main radial potential, A(u), withu = G(m1 + m2)/(c

2 rEOB). Our Eqs. (5.19), (5.20),(5.28) gave the explicit transformation between the pre-viously derived harmonic-coordinates tidal Lagrangiansand their corresponding contributions to the EOB A po-tential. Using this transformation, we could finally ex-plicitly compute the most important tidal contributionsto the EOB A potential to a higher accuracy than hadbeen known before: namely, we computed the quadrupo-lar (ℓ = 2) and octupolar (ℓ = 3) gravito-electric tidalcontributions to 2PN fractional accuracy, i.e., with theinclusion of a relativistic distance-dependent factor ofthe type u2ℓ+2(1 + α1 u + α2 u

2) [see Eqs. (6.6)–(6.10)and (6.18)–(6.22)]. We also computed the quadrupolargravito-magnetic tidal contribution, as well as a newlyintroduced time-differentiated electric quadrupolar tidalterm, to 1PN fractional accuracy [see Eqs. (6.25)–(6.28),(6.30)–(6.33)]. Of most interest among these results isthe obtention of the 2PN coefficient α2e

2 entering thedistance-dependence of the electric quadrupolar term.We found that this coefficient, Eq. (6.10), is always pos-itive and varies between + 3 and + 15.16071 as the massfraction X1 = m1/(m1 + m2) of the considered tidallydeformed body varies between 0 and 1. In the equal-mass case, m1 = m2, i.e. X1 = 1

2 , we found that

α2e2 = 6.07143. This value shows that, when the neu-

tron stars near their contact, 2PN effects are comparable

to 1PN ones. Indeed, contact occurs when the separa-tion r ≃ R1 + R2 = Gm1/(c

2 C1) + Gm2/(c2 C2) (where

CA ≡ GmA/(c2RA), A = 1, 2, are the two compact-

nesses). In the equal-mass case (with C1 = C2), this showsthat, at contact, u = G(m1+m2)/(c

2 r) is approximatelyequal to ucontact ≃ C1. If we consider as typical neutronstar a star of mass 1.4M⊙ and radius 12 km, we expectC1 ∼ 1/6, i.e. ucontact ∼ 1/6. The successive PN con-tributions to the distance-dependent amplification factor

A(2)2PN1 electric(u) = 1+α2e

1 u+α2e2 u2 of the electric quadrupo-

lar tidal interaction for the first body then becomes, atcontact,

A(2)2PN1 electric(ucontact) ≃ 1 + α2e

1 C1 + α2e2 C2

1

∼ 1 +1.25

6+

6.07143

62, (7.1)

where one sees that the 2PN (O(u2)) contribution is nu-merically comparable to the 1PN one. This suggeststhat the PN-expanded form of the tidal amplification fac-

tor A(2)1 electric(u) is slowly converging and could get com-

parable or even larger contributions from higher pow-ers of u (i.e., 3PN and higher terms). In order to geta feeling about the possible origin of this slow conver-gence of the PN expansion, we followed the approach ofRef. [54], i.e., we looked for the existence of a nearbypole (in the complex u plane) within the formal ana-

lytic continuation of the considered function A(2)1 electric(u).

[Ref. [54] considered the energy flux F as a function ofx = (GM Ω/c3)2/3; it pointed out that F (x) had (in thetest-mass limit) a pole at the light-ring value x = 1/3and recommended improving the PN expansion of F (x)(for x < 1/3) by a Pade-type resummation incorporatingthe existence of this pole in F (x).] By computing the ex-

act test-mass limit of the function A(2)1 electric(u), we found

that it formally exhibits a pole located at the light-ringvalue utestmass

LR = 1/3 [see Eq. (6.44)]. Such a pole is alsopresent in other amplification factors [see Eqs. (6.45)–(6.47)], and we discussed its origin. [Note that two equal-mass neutron stars will get in contact before reaching thispole. However the idea here is that the hidden presenceof this pole in the analytical continuation of the func-

tion A(2)1 electric(u) is behind the bad convergence of the

Taylor expansion of this function in powers of u.] Thisled us to suggest that one might get an improved value

of the tidal amplification factor A(2)1 electric(u) by formally

incorporating the presence of this pole in the followingPade-resummed manner:

A(2)1 electric(u) = 1 + α2e

1 u+ α2e2

u2

1− rLR u, (7.2)

where rLR ≡ 1/uLR is the (EOB-defined) dimension-less light-ring radius, i.e., the solution of Eq. (6.55),

with A(u) defined by Eq. (5.21). Let us point out thatEq. (7.2) is equivalent to saying that the 2PN coeffi-cient α2e

2 becomes replaced by the effective distance-dependent coefficient αeff

2 (u) ≡ α2e2 /(1 − rLR u). Note

27

that αeff2 (u) > α2e

2 . In particular, for the “typical” com-pactness C1 = C2 ∼ 1/6 considered above, and when

using the unperturbed value of rLR, i.e. r(0)LR = 3,

the effective value αeff2 (u) will, at contact (i.e. when

u = ucontact ≃ C1 ∼ 1/6), be equal to αeff2 (ucontact) ≃

α2e2 /(1 − 3 C1) ∼ α2e

2 /(1 − 3/6) ∼ 2α2e2 ∼ 12. We re-

called in the Introduction that several comparisons be-tween the analytical (EOB) description of tidal effectsand numerical simulations of tidally interacting binaryneutron stars [5, 9, 10] have suggested the need for sig-

nificant amplification factors A(2)1 electric(u) parametrized

by rather large values of α2e2 . However, up to now, the

numerical results that have been used have been affectedby numerical errors that have not been fully controlled.In particular, in the recent comparisons [9, 10], one didnot have in hand sufficiently many simulations with dif-ferent resolutions for being able to compute and subtractthe finite-resolution error. We hope that a more com-plete analysis will be performed soon (see, in this respect,Refs. [61, 62]). We recommend comparing resolution-extrapolated numerical data to the pole-improved ampli-fication factor of Eq. (7.2). As discussed in Sec. VI, itmight be necessary to use as value of rLR the improvedestimate obtained from the full (tidally modified) valueof the A potential. This suggests (especially for com-pactnesses C1 . 1/6) as discussed above that rLR mightbe significantly larger than 3, thereby further amplifyingthe effective value of α2e

2 during the last stages of theinspiral.

The present study has focused on the 2PN tidal ef-fects in the interaction Hamiltonian. There is also a 2PNtidal effect in the radiation reaction, which has contri-butions from various tidally modified multipolar wave-forms. The tidal contribution to each (circular) multipo-lar gravitational waveform can be parametrized (follow-ing Refs. [5, 10]) as an additional term of the form

htidalℓm (x) =

∑

J

h(J) LOℓm (x) h

(J) tailℓm (x) h

(J) PNℓm (x) , (7.3)

where x ≡ (G(m1 + m2)Ω/c3)2/3; J labels the varioustidal geometrical invariants, such as J2e ≡ Gαβ G

αβ ;

h(J) LOℓm (x) denotes the leading-order (i.e., Newtonian-

order) tidal waveform; h(J) tailℓm (x) the effect of tails

[55, 56] and their resummed EOB form [57]; while

h(J) PNℓm (x) = 1 + β

(Jℓm)1 x+ β

(Jℓm)2 x2 + . . . (7.4)

denotes the effect of higher PN contributions. The 1PN

coefficient β(J2e22)1 is known [7, 15]. The other 1PN co-

efficients needed for deriving a 2PN-accurate flux can beobtained from applying the simple 1PN-accurate formal-ism of Eq. [40]. It is more challenging to compute the

2PN coefficient β(J2e22)2 . Indeed, this requires applying

the 2PN-accurate version [56] of the Blanchet-Damour-Iyer wave-generation formalism [40, 58–60] to the tidal-modified Einstein equations (2.13). Let us, however, notethat although from a PN point of view, the 2PN coef-

ficient β(J2e22)2 contributes to the phasing of coalescing

binaries at the same formal level as the dynamical 2PNcoefficient α2e

2 determined above, it has been found in

Refs. [9, 15] that (if β(J2e22)2 ∼ α2e

2 ) it has a significantlysmaller observable effect.

Let us finally point out that our general result in Eq.(2.20) also opens the possibility of computing the 3PN co-efficient α2e

3 in the PN-expanded amplification factor of

the electric quadrupolar tidal interaction A(2)1 electric(u) =

1 + α2e1 u + α2e

2 u2 + α2e3 u3 + O(u4). This computation

would, however, be much more involved than the calcula-tion of α2e

2 because of the technical subtleties in the reg-ularization of self-field effects at the 3PN level [43, 63–65]that necessitate using dimensional regularization [25, 26]instead of Hadamard regularization.

Acknowledgments

T.D. thanks Gilles Esposito-Farese for useful discus-sions at an early stage of this work. D.B. thanksICRANet for support, and IHES for hospitality duringthe start of this project.

Appendix A: Explicit forms of the (time-symmetric)2PN-accurate brick potentials

The explicit forms of the (time-symmetric) 2PN-accurate brick potentials V , Vi, etc. are [39]

V =Gm1

r1+Gm1

c2

(− (n1v1)

2

2r1+

2v21

r1+Gm2

(− r1

4r312− 5

4r1r12+

r224r1r312

))

+Gm1

c4r1

(3(n1v1)

4

8− 3(n1v1)

2v21

2+ 2v4

1

)

28

+G2m1m2

c4

v21

(3r31

16r512− 37r1

16r312− 1

r1r12− 3r1r

22

16r512+

r22r1r312

)

+v22

(3r31

16r512+

3r116r312

+3

2r1r12− 3r1r

22

16r512+

r222r1r312

)

+(v1v2)

(− 3r31

8r512+

13r18r312

− 3

r1r12+

3r1r22

8r512− r22r1r312

)

+(n12v1)2

(− 15r31

16r512+

57r116r312

+15r1r

22

16r512

)

+(n12v2)2

(− 15r31

16r512− 33r1

16r312+

7

8r1r12+

15r1r22

16r512− 3r22

8r1r312

)

+(n12v1)(n12v2)

(15r318r512

− 9r18r312

− 15r1r22

8r512

)

+(n1v1)(n12v1)

(− 3r21

2r412+

3

4r212+

3r224r412

)+ (n1v2)(n12v1)

(3r214r412

+2

r212

)

+(n1v1)(n12v2)

(3r212r412

+13

4r212− 3r22

4r412

)+ (n1v2)(n12v2)

(− 3r21

4r412− 3

2r212

)

+(n1v1)2

(− r1

8r312+

7

8r1r12− 3r22

8r1r312

)+

(n1v1)(n1v2)r12r312

+G3m2

1m2

c4

(− r31

8r612+

5r18r412

+3

4r1r212+r1r

22

8r612− 5r22

4r1r412

)

+G3m1m

22

c4

(− r31

32r612+

43r116r412

+91

32r1r212− r1r

22

16r612− 23r22

16r1r412+

3r4232r1r612

)+O(6) + 1 ↔ 2 , (A1)

Vi =Gm1v

i1

r1+ ni

12

G2m1m2

c2r212

((n1v1) +

3(n12v12)r12r12

)

+vi1

c2

Gm1

r1

(− (n1v1)

2

2+ v2

1

)+G2m1m2

(− 3r1

4r312+

r224r1r312

− 5

4r1r12

)

+ vi2

G2m1m2r12c2r312

+O(4) + 1 ↔ 2 , (A2)

Wij = δij

(−Gm1v

21

r1− G2m2

1

4r21+G2m1m2

r12S

)+Gm1v

i1v

j1

r1+G2m2

1ni1n

j1

4r21

+ G2m1m2

1

S2

(n

(i1 n

j)2 + 2n

(i1 n

j)12

)− ni

12nj12

(1

S2+

1

r12S

)+O(2) + 1 ↔ 2 , (A3)

Ri = G2m1m2ni12

− (n12v1)

2S

(1

S+

1

r12

)− 2(n2v1)

S2+

3(n2v2)

2S2

+ ni1

G2m2

1(n1v1)

8r21+G2m1m2

S2

(2(n12v1)−

3(n12v2)

2+ 2(n2v1)−

3(n2v2)

2

)

+ vi1

−G

2m21

8r21+G2m1m2

(1

r1r12+

1

2r12S

)− vi

2

G2m1m2

r1r12+O(2) + 1 ↔ 2 , (A4)

X =G2m2

1

8r21

((n1v1)

2 − v21

)+G2m1m2v

21

(1

r1r12+

1

r1S+

1

r12S

)

29

+ G2m1m2

v22

(− 1

r1r12+

1

r1S+

1

r12S

)− (v1v2)

S

(2

r1+

3

2r12

)− (n12v1)

2

S

(1

S+

1

r12

)

− (n12v2)2

S

(1

S+

1

r12

)+

3(n12v1)(n12v2)

2S

(1

S+

1

r12

)+

2(n12v1)(n1v1)

S2

−5(n12v2)(n1v1)

S2− (n1v1)

2

S

(1

S+

1

r1

)+

2(n12v2)(n1v2)

S2

+2(n1v1)(n1v2)

S

(1

S+

1

r1

)− (n1v2)

2

S

(1

S+

1

r1

)− 2(n12v2)(n2v1)

S2

+2(n1v2)(n2v1)

S2− 3(n1v1)(n2v2)

2S2

+G3m3

1

12r31

+ G3m21m2

(1

2r31+

1

16r32+

1

16r21r2− r22

2r21r312

+r32

2r31r312

− r2132r32r

212

− 3

16r2r212+

15r232r21r

212

− r222r31r

212

− r22r31r12

− r21232r21r

32

)+G3m1m

22

(− 1

2r312+

r22r1r312

− 1

2r1r212

)

+O(2) + 1 ↔ 2 . (A5)

Here r1 ≡ x− y1, r1 ≡ |r1|, n1 ≡ r1/r1, r2 ≡ x− y2,etc., y12 ≡ y1 − y2, r12 ≡ |y12|, n12 ≡ y12/r12, v12 ≡v1 − v2, (n12 v1) ≡ n12 · v1. In addition, the notation1 ↔ 2 means adding the terms obtained by exchangingthe particle labels 1 and 2, while the quantity S denotes

the perimeter of the triangle defined by x, y1 and y2,viz.

S ≡ r1 + r2 + r12 . (A6)

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31

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[66] What is important for the current discussion (i.e. theapplication to tidal interactions) is the absence of pre-ferred directions in the orbital dynamics y,v or y,py .This is the case for the reduced tidal actions (generatedby orbital-induced tidal moments).